If you are acne-prone, you should change washcloths and towels very frequently, possibly every day, not just when you feel they are getting "dirty".
My personal "hack" for a similar situation1 was to invest in a whole load of small washcloths/towels (Ikea has packs of ten for 3€ / £3 / $4 and they can be washed very hot and dumped into the drier, for example.). Alternatively, if you can handle a sewing machine, you could cut up and hem some cheap larger towels. Maybe you even have a few extras around? Then wash them together once you have a full load or your stack is running low.
1 In my case, I used them as "wet wipes" during diaper changes and for all those seemingly endles occasions where a quick wipe was in order. Ten years later, they are still used everyday as washcloths and in a few years, once my children hit the "acne age"... Well, see above.