Office supply places have finger cots too. In electronics we use leather thumb guards with an elastic backing  so they stay on.

Can you step into the binder or isn't that possible for you with hips/general body shape?  I find it's near impossible to get mine on over my head because it rolls up as you describe, although I find it hurts my shoulders more than my thumbs.

I could step into it, however...my shoulders and chest is smaller than my hips and belly. I used to pull my binders up but it started stretching them out...especially stretching the arm holes which really was not fun.

Are you putting both arms in at the same time? I find that if I put one arm in, adjust all the rolls out, and then put my other arm in that everything works out fine and it's not difficult to get on.