Tell your future Boss about your visa situation and ask him/her get you offer letter a.s.a.p. and file H1B transfer immediately. After filing, you can start to work for her/him right away.
The key is to get the offer letter. This seems not to be a problem for you. Some schools take a while (2 weeks) to send the offer letter out after verbal. You don not have to be on the payroll of your current school when file H1B transfer. Just use the last pay stub from them should do. Don't worry, everthing will be OK.