1996 Honda Accord Car won't warm up in cold weather.

I assume that the major issue here is the lack of heat to the cabin.

Of course, the easiest thing to check is the thermostat and it is the simple fix usually and the least expensive. This is much better than overheating as that can indicate damage to the engine or the cooling system.

Check the coolant level and see if the Tstat fixes the problem. I don't know how long you have owned the vehicle but I have seen people try to overcome other issues by eliminating the Tstat. Don't do it but it wouldn't surprise me if the Tstat is not there.