New school will need to file a new h1b petition for you. Same process as
first-timers: LCA, offer letter, blah blah. Their lawyer should know it very
well.
The only difference that I know of between a first-time petition and so
called "transfer" is the H1B portability in AC21, which means you can start
working for the new employer once the new petition is filed, instead of
waiting until it is approved. Search "AC21".