You may find this article interesting:
Texas RV Travel Blog on
Black and Gray Water Holding Tank Maintenance.
I don't think you will go wrong on using the bleach. I would put some on a rag and use it in an inconspicuous area to see what the results would be.
The blog linked above tells about water softeners, detergents, and beach for your tanks.
edited: p.s., for the sink stains, I'd use a diluted bleach solution.
TEST TEST TEST - Not responsible for results.
I found the information searching on the words: soap in black tank.