H1B Transfer before stamping

My H1B case is showing as approved, i am changing my employer after its approved from my previous employer.

Is it possible to transfer the H1 now to the new employer? I have got I797C and approval screen shot, Will that be enough from documentation point of view?

Also what is the timeline to get it done through normal and premium process both?

Yes, the new employer can file cap-exempt petition using 797C and print out of online status. Under normal processing, this could take 4-6 months. PP option is always 15 calendar days (unless RFE is issued).