Whichever party takes the USA and imposes conscription into World War I is out of power until the Depression hit. They are out of power forever if you can prevent the Depression, but I think World War I itself caused the Depression so that is unlikely.
By the way, this isn't just true of the USA. Look at the regime in every country in the world, and whichever regime is in power and takes the country into World War I, or in the parliamentary democracies forms the government that takes the country into World War I, goes away at the first opportunity. In the democracies the parties only get a second chance because of the Depression, and not even that for the British Liberals.
So the AHC part is pretty easy. Have a Republican administration, with a post 1917 Hughes administration being your best bet in terms of plausibility, take the USA into World War I. That gets the Democrats in power in both the White House and Congress in the 1920s. They won't prevent the Depression and the Republicans come back in 1932. Just make sure the new President comes from the progressive wing of the party. You can even make it FDR himself without much difficulty.
A clearer left right split between the Democrats and Republicans, with the Republicans on the left though somewhat corporatist, and the Democrats on the right, would have interesting effects on American politics down the road.