If the sponge is too gross to touch, throw it out.

If you have the means, a countertop dishwasher may seem like an expensive item, but it'll use <10% the water of washing by hand, save you from needing to buy gloves every time, and if you're like me, rinse aid will stop your glassware from getting that chalkboard texture. (A refurbished one from a decent company will set you back at most 200, well worth it if you're living alone and not doing a household worth of dishes)