If I have a 100 amp panel. I turn on the stove, furnace, dryer, run hot water, I am now drawing 125 amps or more at the main lugs. Is this an overloaded panel and do I need to upgrade the service?

I would say that you are overloading the panel. I'm surprised the main breaker doesn't trip.

If it were up to me, I would upgrade the service.

And don't wait. The main should be tripping. If it's stuck your overload could cause you a multitude of problems. If you burn up the buss or wiring your change to a 200 amp panel will cost you 2 - 3 times as much. Upgrade when possible.