I tried to use an allen wrench to loosen the little screw under the faucet handle
So I could remove the handle and replace some parts. The faucet does not loosen
At all. What should I try next?

Did the Allen screw come out? If so, spray some WD40 or similar product in the screw hole. If it did not come out, you can try to loosen with WD40 too. Make sure the Allen wrench is the correct one, many faucets now have metric wrenches.