If my bumb shoulder prevents me from drawing back a compound, the a traditional bow is pretty much out of the question, right?

probably because of the no let off

But, a longbow/recurve is smoother to draw and more fluid - no hard break over and holding the draw.

Try one and see - something low weight and shoot some.

I have a friend that tore his rotator cup and he went to a mouth relese on his compound with about a month of practice he was able to shoot very well and he got a elk and a deer this year. he just bites down on this deal and pushes out with his arm