S.C.
We moved into an old drafty farmhouse a few years ago. We don't have oil, ours is all electric. Our first winter here, I couldn't keep the house warmer than 60*, no matter what I did. And our electric bill was about $500 a month during the winter. I think we paid almost $2,000 in heat that winter. It was awful. I would up having to ask for help from HEAP because we simply couldn't afford it. We did a few improvements the following year. We put blown-in insulation in the whole house and installed baseboard heat upstairs. There wasn't any heat or insulation in the upstairs! That made a HUGE difference. The house stayed warmer, but our bill was still crazy high. This year we got a propane fireplace and it's made a world of difference. My electric bill is about half of what it was last year.
If I were you, I'd talk to your landlord about insulating the house. Insulation, especially if it's the blown in kind, settles over time. It needs to be re-done once in awhile. You can hire someone, but if you're handy types, it's not hard to do yourself.
Do you have plastic on the inside of the windows? That makes a HUGE difference for us. You can buy the kits at just about any store. They're easy to do and are fairly inexpensive.