My guess:
High ambient humdity in the bathroom, so the towels & washcloths don't get a chance to dry properly after use. Then over time they grow whatever-it-is that's making them smell bad.
Then you wash & dry them, but the nasties somehow manage to cling to the cloth, and after they're dry, the bad smell is more likely to get into the air just as dust is more likely on a dry street than a damp or wet street.
So yeah, nuke 'em with whatever the folks here recommend, and see what happens.
The bad news is, you might have to nuke the rest of your clothes for a few months to get rid of any residual nasties that transferred to them from the towels & washcloths.
The good news is, whatever it is, probably isn't terribly dangerous since it hasn't made you sick yet. (Look out for athlete's foot and itches & rashes in sensitive areas of the body, in which case keep those areas especially clean, and keep them dry using baby powder.) Though, still a good idea to sanitize everything, because you don't want to go taking chances.
Preventive measure: do whatever possible to reduce humidity in the bathrooms. Including placing a small fan at the bathroom door blowing ambient air in there so the place dries out faster after showers.
