How to tell if shrimp is already cooked and what to do with it

I was always taught to take the head and tail between your fingers of each hand, pull it out straight, push together slightly, then pull apart. That should pull the whole carapace back off the body. Then you can pull off the head (and legs if they're still there) and you're done.

However, I've seldom gotten it to work when raw, and less so when cooked.

I agree, cooked shrimp are harder to peel, and just add an unwanted challenge to the diner when included in most dishes.

Shrimp are extremely lean and have very little connective tissue, which means that the longer they cook the harder they get. Fresh shrimp, for example, only need to be fried for about one minute per side (depending on their size). This makes cooking frozen shrimp very difficult: By the time the outside has thawed and is perfectly cooked, the inside will still likely be frozen. If you cook them long enough for the inside to thaw and cook (e.g., 30 minutes), the outside will likely be extremely hard.

I would recommend thawing the shrimp before cooking, for example, by leaving them in the refrigerator overnight. Another quick way to thaw them is to place them in a bowl and run a stream of cold water over them in your sink. Next, thoroughly dry the shrimp, which will help prevent splatter and promote the Maillard reaction. Only cook the shrimp until they turn slightly pink and begin to curl on one side, then flip. Depending on the size, this should be no more than, say, two minutes per side. Then proceed with the remainder of the recipe. Also take into account if the recipe calls for simmering the shrimp in the cream sauce; if so, then this will continue to cook the shrimp and you will not need to fry them as long.