Am I pulling too many amps?
Yes. Given your service and power demands, the fact you are tripping your main breaker meaning it's doing it's job properly, and your high power bills, so YES, you are overloading your service/system.   I presume your furnace is electric resistance  as well as your clothes dryer.  If so, and you want to stay &quot;all electric&quot; you are going to have to upgrade your main panel to 200 amps. If you have gas available, changing to a gas furnace and/or clothes dryer could possibly avoid the necessity of replacing the main panel.   Its rare for a main breaker to trip, so again, YES you're drawing too much power.  If you don't upgrade, the only thing you can do is time your energy consumption, like turning off the furnace when running the dryer. If you have an electric water heater, you might have to turn it off at times as well.
EDIT:  If you have natural gas available, it's almost always a LOT less expensive than all electric.   If not, propane is another option, but, depending upon where you are and current prices vs. electric, may not save a lot of money on ongoing utility costs.
Checking on your amount of insulation, air leaks and if possible upgrading the insulation and sealing air leaks could minimize your power usage.  But for what you've got going on now, Yeah, you are overloading your service.