Once the truth comes out, the Democratic party will probably split. The Dixiecrats might join the GOP as others have suggested, while the rest might split into (anti-communist) liberals on one hand, and old school progressives/New Dealers on the other. The latter will probably be very marginalized, until they eventually become mostly irrelevant.
However, no matter how crazy things will get in the US, imo they will always remain far ahead of the USSR economically and technologically (unless there‘s all-out civil war or something, which is unlikely). America‘s advantages are simply too great to be nullified just because a kooky government is in charge. Abundant natural resources, a large (well educated) population, world class universities and research institutions, the most fertile soil in the world, a gigantic industrial base, its protected geographic location etc. None of these advantages are going to disappear anytime soon, while the USSR has just fought a war with catastrophic losses - both in human and economic terms - from which it will take decades to recover (and that‘s ignoring the fact that they simply have a far less efficient economic system). Even if they wanted to, the Soviets simply don’t have the economic strength to do stuff like the Marshal Plan for example. Only the US (who made up like 50% of the world economy immediately after the war) is capable of pulling something like that off.
No, the US will retain its advantages, and if they feel truly beleaguered by the Soviets and go full ‚crazy‘, they might even make use of those advantages to a much greater extent ITTL than they ever did during IOTL‘s Cold War. As much as the Soviets and their allies liked to use terms like ‚militarism‘ to describe their enemies, the US has never truly been militaristic at any point in it’s history; if it had been, the Cold War would‘ve probably ended a decade or two earlier.