I think you have to decide what the purpose of preschool is. If you think it's learning sight words and being academically driven, you're narrowing your options. So worrying about them "being behind" is a huge problem. Have you talked to kindergarten teachers about what they want and expect? I'll venture that virtually none of them care about whether kids can read or do math facts or recite state capitals or anything else. They want kids who are secure, can separate from parents, can sit in a circle, follow basic directions, and negotiate their own space without whacking each other. If they can write their name, that's extra. All my colleagues in both private and public schools agree with that - their biggest problems are actually with the kids who are all intellectually "ahead" but don't have basic socialization skills or the level of maturity needed to function in a large class (which may be as much as 2.5 times the size of a preschool class).
I would be concerned if the kids' current teacher may be leaving - I wouldn't blame her based on the pattern you are seeing. I know several have suggested you take her aside and ask her what her plans are, but if I were that teacher, I would not be sharing my professional information with the public especially if there is an issue with management/new ownership. If they treat their people badly, what do you think they will do if they hear this teacher is thinking of leaving? Aside from that, it's really inappropriate for you (or anyone else) to take your professional relationship with a teacher and turn it personal.
Is this center licensed by the state? I assume so. Therefore there are regulations governing things like the toilet - that's a hygiene issue and you say it also contributed to your child's anxiety. The weeds are unsightly and I cannot believe they make the school appealing to prospective families, but they are less of an issue. If it were my school and I used all my money to buy the school and had nothing left for maintenance, I'd be organizing a Family Fun day at the school on a weekend with teams of parents to help spruce things up, see if local nurseries will donate some plants, see if local merchants will donate things for a raffle, and similar things to get everyone to feel good about this school community. But if they're in it for the money only, maybe they aren't thinking that way.
I think you have to look at whether your kids' teacher has enough staying power, or if the school is going to make her decision for her (to save money) and kick her out before the end of the year. You may have to make a decision on someone else's timetable and not on your own anyway.
I also wonder if you have had time to look into other options - are there really only 2 schools to choose between? Again, I'd go for the security and stability of knowing the staff will be stable, and not worry about the academics at all. You say your kids are happy where they are (except for the flare-up of anxiety?) but if you are not proactive about finding an alternative and something major happens (like the teacher leaving or being fired), you'll be stuck. You have the added problem of looking for two slots because you have twins. So your options will be severely limited if this situation goes south without you being prepared.
Good luck.