L1 to H1 while in US and H1 transfer before Oct 1st

I have a unique case. I am on L1B visa in US now and had applied for H1B and it got picked up in lottery.

My question is, if the visa gets approved say on Sept 10th, so can I transfer my H1B to a new employer right away or I need to wait for Oct 1st for my change of status (from L1 to H1), work with the employer for couple of months to get few pay stubs and then initiate a transfer?

Also would like to know what happens if it gets approved on Oct 10th and the start date in I797 is Oct 10th, so can I still work for next 2 weeks with my current employer (on L1B visa) and then start working with my new employer (on H1B visa)?

Any help on this will be very much appreciated!