Yes, you can absolutely get the "Georgia on my Mind" achievement as Fascist/Non-Aligned. I did it myself a few months back with the Loyalists with MacArthur as my head of state.
I think the only USA achievement that requires you to have a democratic government is "Arsenal of Democracy".
The USA achievement "Underpaid, Undersexed, and Under Eisenhower" doesn't technically require you be democratic, but you do have to be in a faction with the UK and have expeditionary forces from them, which is a lot more straightforward if you stay democratic and play historical.