Prednisone absolutely affects the skin. Among other things, it causes thinning that leads to fragile skin that tears and bruises easily. I’m not sure about dark circles and more wrinkles.
One way to improve the wound healing is to take 500 to 1,000 mg of vitamin C every day. It might help with wrinkles, too.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9405326/