Can you delay taking the new apartment for a month?

I'm confused about the "thank you for renewing" email, and the "you are not renewed" statement. Those seem to be contradictory. Make damn sure you have given notice and that you are leaving. Make sure you are not on the hook for another full year.

As for the previous place, you don't have to "sublet" for a month, but just find someone who wants to move in when you are gone, and make an arrangement that they pay you the rent for January. It's a bonus for them, they get the place a month early. There's a fair chance you can get that arranged.