As far as i know the H1B under university jobs falls under non-profit category, so u wont have any quota restriction and application deadlines. But if u wan to join a profit motive organization then u have to file the new application for H1B under the quota (Apirl,1) as per others do. But i hope u can join other universities or not for profit organizations if they are ready to sponsor the H1B or u can take a transfer. As i have heard you have 10 days to land in another job after you have quit the job at one company. Hope you will get other insights from ppl with particular experience.