That's right, pick up the carcass and stow it on your steed after skinning it. You could also lasso the unskinned animal and drag it to the butcher, then skin it after you've sold some stuff. All of that dragging doesn't effect quality. Skinned carcasses can't be lassoed as of a few days ago.
The three star baby alligators are a good payout, far more efficient vs the bigguns since the skins can be stowed like you would deer hides, draped on the back of your horse
It looks like they haven't nerfed the exotic birds north of Saint Denis, thankfully. They have smaller carcasses so you can hang them on the side of your horse, like bunnies.