I would probably leave, but there are a couple of issues here that depend on the laws in your state.
There's no law where I live that says you have to have a men's room and a ladies' room. If it's a single bathroom with a lock, then they can be "co-ed" and in fact, many restaurants here have done that.
The problem comes with the bad habits of everyone else.
I would attempt to force the company to pay for daily maid / cleaning service.
Almost 10 years ago, when I started at my company, our office was similar. Few women (I'm in tech) and our building only had 2 toilets, both with doors and locks. So they were labeled men and women but were really co-ed, and we were required to keep them clean.
We also used to have issues with a few slobs in the kitchen. It drove me nuts to the point of putting up signs to clean up after yourself. Still, a few bad apples would leave a mess, use forks, bowls, and not clean them, leave them in the sink. A couple of us just starting throwing it all away (actually, if I washed something, I'd move it to my desk drawer). It helped a bit but what really helped is when the slobs got laid off.
Of course, we've grown and shrunk and moved now, and I'm quite happy to say that I have a full 4 stalls all to myself. We have 2 buildings and I'm the only woman in mine. Though the other office has 5 or 6 women.
If you cannot get the boss to back you up, it's time to go. If you end up the boss, you make the rules. Guys break them, write them up.