How much to save before moving to NYC?

26m. So at the moment I only have 5000 in my savings. I live in Long Island with my mom and recently got a job in midtown. I'm making $23 an hour 40 hours a week working a CSR position for a company that rents out office spaces. After taxes and deductions from insurance and HSA contributions I'm making $631 a week (I get paid weekly) so basically $32800 a year after everything. I'd really like to move out of my mom's. My mental health has really taken a toll. I feel like I am behind everyone. I want to build a social life and be around more people my age and be able to date since living with your parents at my age is going to be a red flag to a lot of women. If I get a roommate somewhere in the city finding something where I'm paying $1000 a month I feel like I would be able to meet more people and experience more things and build my life as I feel stuck now. If I am frugal I feel like I can have $15000 in my savings when June hits then the plan would be to find a roommate where I am paying $1000 a month. Does this sound like a good plan?.Is that enough to have in savings when moving out? I need some advice I feel lost