A frig that is draining may mean several things: the door seals could be leaking, or, possibly, the defrost cycle is stuck on. Older frigs are often MUCH more energy hogs than the newer ones.
Do you have an electric water heater? Have you checked your water usage? You might have a leak and you're dumping a small, but constant amount of heated water possibly into the ground? Any spots around the foudation that seem wet(ter) than normal?
It's unlikely that the meter is defective, but it does happen.
There are some plug in devices, sort of like an extension cord, that you can use that will tell you how much power whatever is plugged into them is using. One I'm familiar with is Kill A Watt Electricity Usage Monitor. If you do a Google search on that, you can find various versions.
What kind of heating system do you have? If it's a heat pump, and that's not working properly, it may be switching to the emergency resistance strip heaters (i.e., electric heaters), and that will be MUCH less efficient than a properly working heat pump.