Your cat's claws are normal. Cats regularly shed their old claws to reveal strong, sharp ones underneath. What you are thinking is dirt is likely the claw's quick. Also normal. If you trim your cat's claws every week you can trim it right back to just before the dark part. In fact doing that would make it unnecessary to trim the claws that often. When your cat chews on his claws he is trying to facilitate the removal of the claw.
I'd be more interested in why you feel it's necessary to give an indoor only cat a bath every month. You will dry out his/her skin over the long run by doing so. Cats do a great job of keeping themselves clean. Baths are not needed. If, by chance, you are doing it to cut down on allergy suffering by people in the house you would find it much more effective to wipe the cat down every night with a damp, not wet, washcloth. That would remove most of the saliva which, for the most part, is the allergy causing factor.
Assumptions do nothing but make an ass out of u and me.
We can stick our heads in the sand for only so long before it starts choking us. Face it folks. The pet population is bad ALL OVER THE WORLD!