Hmm... well I've done it a couple different ways.
When I just had glass tanks I had problems feeding in the tank. When I would thaw the mice/rats, they would be a little wet and when I put them in the tanks the substrate would get stuck on them! That scared me enough, I didn't want my snakes to swallow the substrate along with the mouse/rat.

Then I switched to moving the snakes to a separate feeding tub. This became a problem too because they got so used to this that I had a couple that would strike at anything that moved. Also a problem because moving them back afterwords was a pain. I was just not comfortable moving the snake back into the tank once it had a full meal in it's belly.

So now what I do is thaw my mice/rats, make sure they are completely dry, which can take a good amount of paper towel. Then I feel comfortable letting the mice/rats touch the bedding. No bedding sticks! It's been working for me ever since.