I usually peel the shrimp. I buy crayfish tails meat only, but my fish do eat whole live baby crays sometimes as they would in the wild, since I think it's a more rounded meal, guts, shell and all. However, it's probably safer to feed without shell, and while I know what risk I'm taking with my own fish, it's up to you to decide that on your own, so I'm not going to recommend feeding with shell.
If you want to minimize the risk of a hard shell, you can always keep some crays and feed them to the fish right after they molt when they're soft.
I don't recommend feeding anything with a shell other than every once in awhile and in a very small amount either way. They need to be small enough for the fish to be able to digest and pass easily.
You can gutload crays with flake and pellet, and you can do the same with crickets or other bugs.
For the average person I'd suggest frozen over live, other than for bugs and earthworms. It's easier, not to mention you avoid many parasites and bacteria by using frozen.