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Sorry it took so long to update you on the issue, but I haven't had a day off work since October 2nd and just didn't have time to work with the breakers or wires.
I did get the issue resolved. There was a bad breaker which I did replace, and I found the open ground to an old model baseboard heater. The heater doesn't work and was on my list to replace, so I just removed it and disconnected it from the breaker box completely. I doubt that had anything to do with the main issue, but figured it couldn't hurt.
After that, I really couldn't get the problem to repeat no matter what I tried, so I think the problem is resolved. But I did go ahead and put the furnace on its own dedicated breaker anyway.
Thank you so much for all the information. I would have never figured this out on my own and was getting quite frustrated.
I am very happy with the help you gave. Thank you again.