I started skipping dish cloths altogether and just went to Handi Wipes. Problem solved. Crazy, I know, but you gotta do what you gotta do! Never had a problem after I started using these. If you've never used them, they work just as well as regular dish cloths, and yes they are washable. And cheap, too! The only issue I ever ran into with them is if kids threw them in the wash with little clumps of yeast dough stuck to them - then they'd have to be thrown away, because they'd get all gnarled up in the wash. But cheap to replace! By the way, I highly recommend using bar mop towels. They're wonderfully absorbent and students won't need to use as many for each lab.
Related: for you newbies out there, you can NEVER have too many dish towels and dish cloths! If you have parents* who want to make donations, a little extra money in your budget (hahahahaha), or you're given a small WalMart or Target gift card and aren't sure what the best investment is, go for towels!
*Or fellow teachers/staff members. I worked with a lady once who asked what kind of supplies we needed for the kitchens because she loves to shop over the summer and doesn't really have people to buy for anymore now that her grandchildren are grown. Don't ask questions - just ask for towels!