House is not airtight and the heater suffers

My house is not airtight. I realized that along the baseboards the workers skimped a bit. They used insulation in the walls, and sort of loosely did it along the floor. Then the wood baseboards hid the shoddy workmanship. I can feel that they did a poor job though. The cold air is right around my feet level and it is horrible. I always have to wear socks and sometimes slippers in the winter. If the issue was my door frame or windows leaking in air, the fix would be simple. With the baseboards being the issue, I would have to do some major construction. I would need to tear out the baseboards, part of the wall and add in insulation. I really don’t want to do this. However, I think I might need to. My heating bills are obnoxiously high. It makes sense though. The heater is set to heat my home until it is around 70 degrees. When most of the heating is leaking out of the house and the cold air is leaking in, it can’t achieve the thermostat temperature. The heater then just keeps heating and heating. Not only is this bad for the energy bills, but I am worried that my heater will overwork itself. The heater could possibly overheat and quit. Major heater parts can get worn out doing this. So maybe the work on the baseboards are not as bad as I think they are. The expense of the repair vs the furnace bills should be comparable. Also, I will then have quality heating.

heating provider