<human>: **BIG NOTE**: I am not looking for binding legal advice, just information!

So the story goes like this:

Wife and I are unsure if we are going to move, so we decided to go month-to-month on our lease from July 1 2011 until October 2011, which was $100 extra on top of our regular rent. 

Come October, we get news that we'll be in the area another year (work related), so I call the leasing office and sign a new lease on October 9th, but effective October 1. My rent check had already been sent for the month (automatically done via my bank), so I ignore it and then forget to change my check, so the November rent also goes out with an extra $100 in it. I promptly change the check for December and keep reminding myself to call the office about this.

So time marches on and today I get a phone call. One of the office workers wanted to let me know that I have a $100 credit on my account, so I can just make my next check out for that much less. I bring up the topic of the $100 at the beginning of October, and she says she'll have to check on it, as she thinks that because the rent was due that day, even though the new contract was signed, it's still valid and they don't owe me the other $100. 

I was very cordial and expressed that I wasn't trying to be greedy but I just want to have balanced books, but her reasoning isn't really sitting straight with me. She was going to call me back but I never got a call, so I'm expecting one tomorrow. 

Any advice or cases I can look at that would point me in the right direction? I'm fairly laid back about money, often letting little things like this slip, but I could use the extra $100.

FYI, we are in NJ for any applicable case law.

Summarize the given text in a few sentences.
<bot>: Overpaid two months on rent, signed a new lease after sending a check and wondering if I'm entitled to a refund for the difference based on date contract was signed vs due date of rent.