yes, after graduation school gives out new I-20 dating till end of your OPT date. But just to be in secure side, its better to apply for H1 from your school and then apply for new visa. There are some possiblility that they might not issue visa based upon  the job offer( from your school) which will last  till OPT is valid( " Jb will last for almost a year").

H1B visas for non profit organization should not take more than a week to be issued. My roommate got his issued within 3 businessday.

Its your call. the ball is in your court ,you can play it safe or go " highway or myway".

Good luck.