Can I ask my new employer to file H1B w/COS for me with start date Dec 1st 2012?. I know typically start date is Oct 1st. I am on L1B now and I can’t leave my current organisation till end of November.
If the COS is already applied and gets approved, then you will be on H-1 from that COS effective date. There are things you can do to invalidate the COS (like traveling before COS approval or returning on different visa stamp soon after COS goes into effect). But if you don’t take any such action, you will be considered on the new status.
If COS is not applied, then you remain on old status. To move to new status, employer can file a separate COS later, or you can enter on stamped H-1 visa.