I know that insulin has messed with my internal thermometer. I get ridiculously hot and sweaty from basic activities (I am otherwise perfectly fine, no diabetes complications) and I can't sit in the sun for more than 5 minutes before I just boil up.
My best advice is to speak to your GP to rule out any underlying conditions, and in the meantime try to stay indoors between 1-4pm as much as possible. The UV rays in the shade are still very strong and you can get sunburn and rashes from the shade, especially when the sun is hottest during the day. Stay extra hydrated and generously use lotions with aloe, coconut and/or shea butter.
Dry and sensitive skin is common in diabetics, particularly type 1 (eczema is often related to auto-immune conditions, of which type 1 is a part of). Not much we can really do about it other than using adequate products and taking preventive measures!