I'm no expert so take what I say with a grain of salt.
If the coat hanger and tin can tricks don't completely solve it here are some other thoughts.
1.If your dials have words, ignore them and only pay attention to the letters. The large burner on my electric stove will keep a large pot simmering between warm and warm. Usually I'll bring to an almost boil on med and then turn down the heat or if it's critical, move the pot to another burner preheated to a lower temperature.
2. As for pots. My experience is that while I can rarely afford the most expensive anything, it’s never more economical to buy the cheapest. For about the cost of one All-Clad sauce pan you could buy a set of non-stick Wear Ever heavy disc bottomed pots and pans with a lifetime warranty. Which of those would be the better choice is arguable but the $1.99 pan or the $19 set is not, at least in my opinion. If I was a 16 yr old with a serious interest in cooking, those All-Clad pieces (one at a time unless I had a rich uncle) would be on my Christmas list.
3. You don't mention what kind of sauce is burning. For a hollandaise, for instance, you might use a glass or stainless steel mixing bowl on top of your pot as a double boiler. This would trump both the electric stove and pot problem.
4. My stock has outgrown the pot. I used to use the 5-quart Dutch oven that came with the Wear Ever set mention above. Thanks to the corrupting influence of all these gourmands, last time I used an 8-quart stockpot I originally bought for 2.99 to use as a trash can. Pots don't get any chincier than this one, but even after more than 6 hours, while it might have overcooked, it didn't scorch or burn on an electric stovetop. If I haven't replaced it before the next time I make stock, to buy a little insurance, I'll use a cast iron griddle/Dutch oven lid as a heat diffuser.