Fire under electric stovetop element

As long as you have a dutch oven and are planning on using it, look for camping recipes for dutch ovens. Usually when camping the suggestion is to put a certain number of coals under the dutch oven and a certain number on top to basically simulate oven-like conditions. You can make warm breakfasts and warm dinners this way. You can even bake bread and other baked goods in a dutch oven (Irish soda bread works very well in one).

Other options for cooking in the fire include wrapping food in foil packets. For example you can wrap up salmon with seasonings or a packet of beef and vegetables. Foil packet dinners will be quicker and easier than most dutch oven dinners but the heat control is not as precise.

Another option if you want to invest in some gear is to get a pie iron. A pie iron is basically two sandwich bread shaped pans that clamp together. You can get one in cast iron, which will hold up longest. Using a pie iron you can make cake, cornbread, and all sorts of warm things sandwiched between toast.

Finally if you have some sort of grate that goes over your fire you can also attempt foods similar to grilling.

With your existing tools you can make some fine meals. With a few more purchases you can make enough food variety to keep you happy for an entire winter of power outages - on the dinner front at least.

I don't think it really matters exactly what happened. Unless you've left out something important, it's not working right, and you need to get it repaired. The best case is that it keeps burning your food, and the worst case is that you set your place on fire.

If you want to confirm that it's doing something wrong, you could put a thermometer in it and turn it on (no food) and confirm that it's getting way too hot. (Edit: but please be careful, as Aaronut said. If there's any sign things are going wrong, turn it off, and leave the door closed.)