It is never a good idea to stick around with someone who cannot give you a straight answer about your relationship status, and who runs hot and cold. Rest assured people like that want to keep you on reserve and don't actually care about your welfare or your relationship. It is also never a good idea to become involved with a person when your involvement could cause them trouble at work. It sounds like you have experienced some drama at your school on account of your involvement, which only proves this point, and I think it will only get worse over time, especially based on the responses you have received from management. Students should attend to learn and nothing else. Don't mix personal with professional. Thirdly, it is a bad idea to hang out with people who say to be careful of whom you trust (in this context). This is an extremely negative and harmful mindset that will impact your ability to interact with others in a meaningful way in the future. For your sake, discontinue this involvement.
Run, don't walk, away from this person.
Why do you even want to stay at a dysfunctional place like this? Management's sexist comment about "how bad you looked as a woman" should be a turn-off. It's not about your sex. It's about two people who made a mistake, and management should be acknowledging the other person's role in it too, more so than yours if he is an instructor or your senior.