Your broker doesn't have an execution desk? They should, your FCM should. Get this filled down in the pit, they will get you a more or less fair fill. Or just offer at today settlement minus a few ticks on Monday. Someone will want the free money.
""F likely has more liquidity and gives you more time to get out." How does it give me more time? I have a deadline of Monday's close to get out or they'll get me out the next day." They will give you more time to get out of the January position because it doesn't expire next week.