Now, as for the rooster... you can get these to help with that
https://www.amazon.com/TOBILE-Chicken-Feather-Protector-Elastic/dp/B0BZLH4WGP/ref=sr_1_8?crid=1W9XIZBH7M4PO&keywords=chicken+saddle+for+hens&qid=1692895337&sprefix=chicken+saddle%2Caps%2C127&sr=8-8
It will protect their backs, but...
Dont worry to much about the cold. Chicken do MUCH better with the cold then the do the heat. With this heat wave, mine are suffering here! Just make sure they have a well ventilated, draft free coop.... they will be feathered out in no time!
The molting season is just starting.... They may be new feathers coming in. Dont pull anything, just leave them be. If they are meant to come off, they will.