L1B to H1B COS, Change companies without Stamping?

[kicknuts]

Hi,
I was originally on L1B Visa, I won the lottery and got H1B approved with same employer. My lawyer says that to activate status to H1B I will have to
get a stamping. Now I am looking for changing my job, so can my new employer transfer this H1B? So should I exit and re-enter
before I can change my job?

[kicknuts]

Can anyone answer this? Is it possible for approved H1B to get transferred to another employer while I am on L1B with current employer?
My H1B was filed as consular processing (form B).