I would think you could process them almost any way you like. You could still dry them for sure. They will just be a bit softer and mushy when you finally do eat them but if they are not rotten, they should be fine....I would think. I had some Hot Hungarian Wax peppers get soft before I picked them but after slicing and canning them, I cannot tell any difference between them and the firmer ones.

I wouldn't can anything that was "squishy". I did have some bell peppers that were getting a little wrinkled and soft, as well as some chiles that were just a teeny bit wrinkled, not smooth and crisp enough to pickle. I made jelly from them, didn't use the peppers themselves, just the juice, and didn't use the most wrinkled/soft parts of the bells, just the unwrinkled parts (though I figured slight wrinkling of the jals was OK for the ones I pureed - used firm ones for mincing to add to the pot later).

I plan on freezing the jals and serranos (as well as larger Ancho types) that are getting wrinkly. FWIW, pulling the plants and hanging them in the basement didn't really help them ripen - or at least, they didn't stay crisp while ripening.

The cayennes and thin-skinned/small peppers (like Thai) are going to be dehydrated.

Here is a link that might be useful: Shriveled pepper thread

I'd dry them for powder.
Then use them for additions to other stuff-Chile,Rubs,sauces etc.

You could also smoke them and have chipole.
John A

yep, smoke them and/or dry them.