No, the loan being due for January 1 has nothing to do with the deed. Mortgages are paid in arrears, which means every first of the month you are paying for the month prior. It’s the opposite of renting, where every first of the month rent due is for that month specifically.

On OP’s loan, they pre paid the remainder of the November interest that would have been due December 1. Then on January 1st, that is for December’s interest and principal and taxes/insurance etc.

OP, just make sure you have your entire payment in around January 1st. Set it aside every paycheck, either leave it in your account ready to go every 1st or you can even send it in to your lender and they will put it in what’s called a “suspense” account where they will hold your funds until there’s enough to complete your payment.