I'm unclear why you are getting pushback about the adjunct job. Is this a situation where students leave at 3, but you are contracted to be there til 3:30 and you are trying to leave at 3:05 to get to the adjunct job on time? In that case, no, you can't do that. You have to stay until you are contractually obligated to stay.
But I understand your feelings on the matter. I moved to a private school where I have a full time TA, and because of his help (and because this school has less pencil and paper work, less testing) I literally do NOTHING outside of contract hours. I leave as soon as the kids leave. It is very freeing and after decades of giving my time away for free, I now use my time only for me and my family. I also used to spend time after hours prepping things because my students came to me so severely behind (poverty, ESL, etc). I felt I could make a difference for them, and did, but a huge personal cost. Teaching very wealthy kids can be freeing because I know their families have money to pay for private tutoring if they need it. I do my best work during contract hours, but give nothing more.