I don't know the laws regarding this, BUT I would say in writing tell the landlord you are surrendering your apt. on the 16th with your final payment of $$$ to cover the rent will be or was made on such and such date--have them sign it agreeing to it. Then you would be done and not liable. Be sure to discontinue any utilities you were paying, mail forward etc. Again I am not giving you legal advice, just my thinking of what might help. I think (?) since your agreement was month to month and you are paying the last month your obligation has been filled and now ends and they can rent the place out.

The landlord legally can't get rent from both you and the new tenant. You should get some money back from the landlord for the period Dec 20-31.

There is no way that this is supposed to be like that, it's most likely illegal and would recommend telling him/her that this isn't something he can just do like that.

If you've paid rent for the whole month you have the right to stay until the end of it or get the proportional amount of money back. Also, I always prefer to have some sort of agreement in written even if it's not a proper contract in order not to have bad surprises when things like this happen!

OP has moved on, but they should have agreed on exact date his lease terminated (end of the day on the 19th) and pro-rated the rent between the two tenants. There's this thing called a handheld calculator - the landlord could easily have figured what 19/31 of the rent was & charged OP only that amount.

yes, that it's illegal to collect rent twice.