If you are looking to upgrade accuracy, I would not trade an M1A for an FAL. Both are excellent rifles. In my humble opinion and those of a number of others, the M1A is the better platform upon which to build an accurate rifle. Start with headspace and the normal upgrades. Unless you have a really jinxed rifle, the M1A should get you where you want to go within the reasonable limits and what your pocketbook allows.
I own a number of FALs. Some are of respectable accuracy, one is excellent, one is poor. Shooting from a bench, I would go with the M1A. Much better sights, fixed sight plane, "tweakable" gas system, far superior trigger group, can be bedded instead of floated (fore end), good track record in matches.
FALs are excellent rifles and can be very reliable, but the M1A has an edge on accuracy.
Why not build a FAL, to try out? They are relatively inexpensive to build and mags are dirt cheap. For many, they have excellent ergonomics.
But I would not trade off an M1A for an FAL, unless the M1A is a real dog.
Just my $0.02