Job Claimed They Had "no Positions Available" After They Accepted Me And Made Me Quit My Former Job

Location: Florida

So a little backstory, at my old job I was a full time and I was trying to find a part time job because I was going back to college. I found a part time job that fit my schedule so I applied to that job. A week later the part time job called me and told me they wanted to schedule an interview, after going through the interview process I was accepted by one of their managers and when they asked me how soon would I be able to start I told them I had to put in my 2 weeks from my full time job first. They agreed and scheduled an orientation in 2 weeks to allow me to put in my job resignation which me and the hiring manager then shook hands on. After I put in my resignation, a week later the part time job called me and asked me if I was interested in a seasonal or full time position because they don't have any part time positions available. I told them I wasn't interested in a seasonal or full time job and I had specifically applied and got accepted for the part time job. Now they're saying they have to talk to the manager again because they have 0 permanent part time positions available.

I'm wondering if I can escalate this and sue for financial damages since they made me put in my 2 weeks for my full time job which will leave me jobless if they don't figure out the part time position situation. The manager even shook hands and reverified my start date for the position and we even went over compensation. I don't think this is right and would like advice on any legal actions I can take.

Thank you.

EDIT: I was looking into Promissory Estoppel which is a legal principle in Florida. I'm wondering if I could get any leverage with this?