Hmm, well, it may or may not change the attitude of Constantine VIII. On the one hand, having a bond with a major family of the military aristocracy of Asia Minor could perhaps allow him to feel a bit more secure on his throne, which means that he would probably adopt a policy of less liberal use of the pwers of the state to stamp out any real or perceived threat to his person and power. On the other hand, his advisors, fearing for their influence, could perhaps make him suspect his son-in-law more and thus make him even more paranoid than OTL (especially since he would be his successor). Assuming that things proceed according to the first scenario, then by 1028/1029, Constantine VIII would be dead and Constantine Dalassenos would be on the throne as Constantine IX.
Domestic policies: Since Constantine VIII wouldn't have allowed much meddling from his son-in-law, he would have most likely imposed the same or similar fiscal policies to the OTL ones. This means that Constantine IX would have to deal with the repercussions of these measures: unrest in Greece, the flight of small landowners and farmers and the abandonment of their property and the consequent increase in the tax burden of the wealthier persons of their communities (the
allelengyon). In that area, I think he would adopt pretty much the same policies as Romanos III Argyros did: they come from a similar backround and have similar interests. Therefore, he would probably adopt measures to support the return of the farmers to their areas of origin and loosen the system of tax and fiscal controls put in place by Basil II (a sort of tax holiday). He could perhaps proceed with abolishing the
allelengyon, although, unlike Romanos Argyros, he already had a good powerbase to start with and is a member of the Anatolian aristocracy, therefore he might think that he doesn't have to go at great lengths to secure the support of the latter; thus, he mightjust reorganise the system and make it less burdensome for the aristocracy instead of outrght abolishing it. For similar reasons, he might not resort to the policy of large-scale benefits and handouts to the clergy and the population of Constantinople. All these combined would help keep the empire's finances sustainable and in good shape a bit longer than OTL instead of the beginning of the finances' deterioration that started IOTL with Romanos (not so much perhaps if he decides to abolish the
allelengyon).
Foreign Policy: It is here that Constantine IX would be more "original". IOTL, Romanos III, partly out of personal ambition, partly out of a desire to prove himself as an accomplished military leader in order to consolidate his position (especially given the lack of any mlitary record of his), had grand designs on the East (according to sources, he contemplated the conquest of Syria and Egypt). However, his inexperience resulted in various serious troubles: the campaign on Aleppo in the summer of 1030 ended up in disaster and the naval campaign against Egypt three years later was met with a similar fate.
Constantine on the other hand has a long experience on the affairs of the eastern border, having served as duke of Antioch. As a result, he would most likely adopt a more careful policy in that area: he would probably be in favour of minor readjustments to the border (such as the conquest of Edessa); he would also continue the policy of maintaining suzerainty over Aleppo and propping it up as a buffer state between the empire and the Fatimids. Therefore, the empire would probably be in a better position in Syria compared to OTL. This doesn't mean however that he would remain inactive. There is a good possibility of him deciding to launch a campaign to capture Sicily (after all, it was kind of unfinished business, since Basil II planned the same just before he died).
Now, the interesting question is whether he would command the troops and lead the campaign himself. On the one hand, he doesn't need to prove himself as a military commander that much, unlike Romanos, the campaign isn't that important that would demand his presence on the field and he is quite old too (even if he was born in 970, he would be by his late 50s - early 60s by the time he would have become emperor); on the other hand, he has military experience and perhaps he too would like a victory of his own to further reinforce his position; personally, I think that he would have remaind at Constantinople. Assuming that he entrusted command of the army to an at least somewhat competent military commander, he could perhaps succeed in wrestling control of the island from the African emirs. Success in this operation could perhaps result in a more energetic policy with regard to the West and more efforts to prop up imperial rule in southern Italy (perhaps some early operations against the Normans?).
Dynastic affairs: Thus, if all these happened, Constantine would probably leave the empire in a better shape than Romanos did. The better state of the state finances would probably preclude the urgent need for a competent financial administrator that could use that situation to amass personal influence, such as John the Orphanotrophos. Furthermore, the better financial situation could perhaps allow the size of the court expenditures to remain unchanged, which would mean that Zoe's spending wouldn't be curtailed, which in turn could allow the couple to remain on better terms.
This however doesn't mean that all would be well. Theodora would still be around and she would most likely still have designs on the throne. She could therefore become the centre of the opposition ITTL as well. There would still be trouble with various groups: the bureaucracy, who would see their way to greater power and influence, already paved during the reign of Constantine VIII, blocked, and fearful of the possibility of the return of the military aristocracy, they could side with Theodora; the Constantinopolitan aristocracy could also become another group to support Theodora, worrying about the possibility of the large Anatolian families once again dominating the state; and perhaps parts of the clergy and the more affluent Constantinopolitans could side with Theodora, because of Constantine IX's more conservative fiscal policies. Therefore, we could see the gradual formation of two camps.
This doesn't mean thought that Theodora would make any attempts. Constantine Dalassenos would be in a stronger position than Romanos (especially if the empire's economic well-being has been restored and the emperor has scored some victories), which means that he would be more willing to adopt harsher measures to deal with Theodora if she stepped out of the line, while the aforementioned greater popularity could perhaps dissuade Zoe's sister from making any moves. If she did move against him, then I think that he would have little trouble with confining her into a monastery (even IOTL, after all the setbacks, Romanos was able to force her to become a nun and it didn't cause that much of an uproar, most likely because, in the minds of the people, Zoe was still around to continue the Macedonian dynasty).
Succession would be tricky: ITTL, there woudn't be a situation where the empress is prepared and she can marry (regardless of the conditions) soon after her husband is dead. Assuming that the marriage between Zoe and Constantine wouldn't have resulted in any children, the throne is up for grabs. Constantine IX could probably try to promote a relative of his as his successor, which would cause problems since Zoe wouldn't like to be sidelined and the population was still fond of the Macedonian dynasty. A compromise could perhaps be that Zoe would marry another member of the Dalassenoi family, in order to satisfy both the empress and the family of the deceased emperor. However, this would still be far from perfect: other Anatolian aristocratic families could start resenting the arrangement, which would continue the presence of the Dalassenoi on the imperial throne, to their potential exclusion. With some concessions though to the aristocracy (tax exemptions and the abolition of the
allelengyon), the Church (donations, tax exemptions) and the affluent Constantinopolitan population (donations, the offer of titles and the opening of offices), the Dalassenoi could perhaps secure enough support (or tolerance) to get a second one of their own wear the imperial purple. In that case, we would probaby see a continuation of the policies enacted by Constantine IX
There is also the possibility that Zoe could try to outmaneuver the relatives of her now dead husband: perhaps by using the divisions between the Dalassenoi and other aristocratic families (if she played her cards properly and had some good advice), by pitting the two sides against each other, she could gain enough freedom of movement to pick a more pliable consort and make moves to secure support. She would therefore probably turn to the Constantinopolitan aristocracy to find a husband, since she could expect that a consort from those quarters would rely more on her for legitimacy and would thus be less threatening to her interests.
However, there is also a third possibility: Theodora. The sister of Zoe has actually a good chance to claim the throne for herself: if she has bided her time, creating alliances in the bureaucracy, the aristocracy and the Church, she could perhaps succeed in outmaneuvering her sister and marry quickly enough, thus being ready to claim the throne. Her victory could perhaps mark the return of the more "civilian" aristocracy/upper class to greater prominence; relations with Zoe would be rather frosty, since Zoe would still technically be empress. If Theodora attempted to neutralise her by putting her in a monastery, then there is a good chance Zoe would flip, which could have serious consequences, especially if various quarters decided to use this as an opportunity to advance their interests.
If Theodora managed to consolidate her hold on the throne and remain influential in decision-making (she seems to have been more interesting to the exercise rather than the trappings of power, unlike Zoe), then we would probably expect the civilian aristocracy and the bureaucracy to grow in influence. The aristocracy would almost certainly receive some concessions in exchange for their acquiescence, most likely in the form of lower taxes. The Church and the Constantinopolitan middle and upper classes would also receive some benefits from the imperial government. But it would probably be on a rather smaller scale compared to OTL, while the relative fiscal prudence of Constantine IX would mean that the empire's finances aren't problematic. There would still be tension between civilian and military leaders though.
(hope this is good enough and without any mistakes
)