Advice needed- urgent

So my OPT is ending mid June, and I am currently working in an unpaid role. I got a verbal offer from a potential employer for a paid role. But it might take a while to get a written offer letter. I want to push them to expedite the process but here is where I am confused:
Do I need to start working for them before my OPT ends or can I get an offer letter for a joining date that is later in June and just apply to STEM before my OPT expires with just the offer letter as proof