Seriously, why are you refusing to check with your HR? Assuming you need to stay till 31 Jan, you CANNOT join your new firm until 1 Feb. You can't use your leave to extend your service in current employment and also work for another firm during the same period. Since you're leaving, just check with your HR and they will lay out all options for you.
My personal advice is just forget about the AWS. It's just but 1 month salary. Your career will be ~480 months, why stress just for 1 month salary? Go in your new firm earlier, the earlier start will more likely to give you more monetary benefits over the stretch of your career.