I would not try to patch the sinks anymore and refinishing is expensive. The cost to refinish a bathroom sink would far outweigh the value obtained, unless their was no other suitable replacement available and/or you were dealing with an historic remodel. If your sinks have vanity cabinets beneath them, you can purchase cultured marble tops that have an integrated sink and give your bathrooms a new, fresh look with only minimal outlay of cash. The tops come in standard sizes and can be easily cut to fit the space, if necessary. It may be time to replace your vanities also, since you mention a suitable replacement sink could not be located. For $200 - $400 dollars, both bathrooms could have new sinks, faucets, and vanity cabinets, and refinishing would be at least $200. Good Luck.