Travel to India with EAD card and new employer offer letter

Hey guys,
I have a situation, I am currently under an employer and I have an offer with a new employer and it starts in March.
First question : Can I work for the current employer after accepting the offer with new employer?
Second question : I will apply for F1- STEM now and my F1 OPT expired on Feb 3rd 2021. If I stopped working with current employer on Jan 31st 2021 then I have 1 month time period, can I go to India and come back without any restrictions?

Please let me know.

Yes, you can. On OPT, you are fine.
Well, traveling out of country on OPT can be tricky. There are many cases, that are successful, there are some cases where there were issues too…So, it is grey area. Talk to your DSO and your new company attorney before you travel.