Drain my FSA right before quitting and shortly after the new year?

I'm planning on quitting my job in February or March, but I'm wondering what to do with the funds in my Flex Spending Account? It's currently empty (because of medical expenses a few months ago) , but on January 1st, I can choose to put up to $2500 into my FSA immediately (which just means they will take ~$200/mo out of my paycheck the rest of the year). But I still have a bunch of medical bills that I owe money on and could use all of that $2500 in January. If do that, and then quit a month later, is my employer going to expect me to pay back the difference (~$2300) since I technically didn't contribute more than a month's worth?

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