Is my employer to pay me if I put a 90 day notice in and they want to let me go before

I have been working for a big bank for 3 years now. This bank has a harassment policy which includes sexual orientation. I have had a couple of issues with employees making comments that I have reported to HR and my managers. I basically had to convince them to address it. Now I am being retaliated against and was told by my AVP that I am the problem and that I will get penalized on my next review if I am not social with the same people who have not only created a hostile environment but said these hurtful things to me. I also got yelled at by my supervisor in front of everyone. I advised her that I wasnt going to allow her to speak to me that way and that we could take this behind closed doors. This escalated into me calling her out on her bullying me. She has treated me different than the others on the team. Everyone can see it. This was escalated to senior mngt and they are refusing to address this issue. I have asked to be moved to another manager, another team, and out of the dept. All of these requests have been declined. At this point, I feel that my only option is to leave the company because there is no fixing this. I have advised my seniors that I have suffered depression because of the treatment and they dont care. My question is this. I have a job that I can start on November 1st. Thats three months from now. Should I submit a 90 day notice and advise them that I have another job and would appreciate them letting me work out the whole notice? If they ask me to leave before the 90 days, are they required to pay for it? I have advised them that I take care of my aging mother, and money is tight. I cant go without a job. In the end, I am sure that they are getting what they want, which is me leaving. Should I make a request through HR? Would it make a difference?