As for dropping my bow arm, it's not so much an issue after the shot, whereas when I'm actually aiming. I will line up my pin and try letting it float over the spot, and almost immediately my arm begins to drop, and then I have to fight it back upwards to get it to stay and when I do this, it leads to fatigue and my arm begins to shake even more causing my pin to float much much more.
This, my friend, is what's referred to as the dreaded TP... unfortunately, I doubt switching bow arms is going to cure that in and of itself... Learning to achieve a surprise release, with back tension, is what's needed here -- along with plenty of time put in on aiming drills without the act of firing the bow.