Kitten keeps trying to eat rug and wont use scratcher

I have a pretty new kitten, she's 5 months old. I got a new rug last week and she keeps biting and eating and scratching it. Its a decent rug, brand new, and I don't want her to do that. When I try to get her to stop, she scratches me. Im usually fine with it but its very cold and dry and my hands can't take it right now. She has a cat tree with scratchers and a scratching pad but she doesnt use either. Her claws are really sharp because of this and make it really painful/bloody when she scratches me. I ordered a spray bottle online because I don't know how to get her to stop without getting scratched.

Does anyone have advice on what to do? I don't want to be mad at her but she really hurts me sometimes and wont listen. Ive never had a cat before so I could use some discipline tips or tips on how to get her to use the damn scratchers