For your situation, never ask

your company whether you can transfer. Since they told you that it is legal for you to work for your company after 12/1, then I am sure you can also transfer to another company. If you don't want to have to worst scenario of being laid off, it is better for you to talk to your new employer to submit your H1B transfer as soon as possible. Finally, you can choose not to go to the new company if you can stay safely and more comfortably with your old company.

Good luck, and no nee to thank:)