Suddenly, John Quincy Adams gets the crazy idea that Andrew Jackson might be a political threat to him. He joins the calls of others in the Monroe administration, and Jackson is censured. Adams-Onís Treaty goes as per OTL, but everyone mad about the borders is even MORE mad about the borders. If only they hadn't forced Jackson to quit, then we have Texas.
Andrew Jackson likes people who like Andrew Jackson, so he heads south and west, along with his friend Sam Houston and nephew John Coffee Hays. Meanwhile, citizens outraged at the treaty are begging to mass in Natchez, Mississippi. After talking to the very persuasive Doctor James Long, Jackson rides into the Natchez camp is overwhelmingly elected leader.
Jackson and four thousand men roll into Spanish Texas and raise hell. Jean Lafitte has seen Jackson beat the Spanish before, and it looks like he's going to do it again. So he pulls a double-cross and joins the rebellion. With more men (and others trickling in daily), more supplies, and more support from the US population, Jackson and friends actually pull it off. In the first election of the Republic of Texas, Jackson runs to replace Acting President Long. He's the only candidate.
After two six year terms, in 1832 Jackson is replaced by his close friend and protege, Houston (no Eliza Allen, no Indian years for Sam). Sam's still in his first term when Mexican President Santa Anna (fuck, how does he always end up in charge?) attacks Texas, looking for an easy foreign punching bag to raise popularity at home.
It fails horribly! Santa Anna comes away claiming the treaty that ended the war was forced, but even if Mexico wants to reclaim its ceded territory (OTL's Mexican Cession, Gadsen, and Baja), it rapidly collapses into an increasingly slapstick series of civil wars. Texas fills up with settlers, especially after the US's shitty civil war starts in 1856 when the south realizes this Missouri Compromise isn't working out so well for them. Texas's lasting love affair with slavery makes it a bit of an international pariah, but more South Africa in 1985 level than North Korea 2012 level. Then the oil boom rolls around - and since the Republic hasn't bothered to change that old Spanish law that gives the sovereign state full ownership of all oil and minerals, they do alright. By 1948, Houston and Jacksonville (San Francisco) are the two largest cities in North America.