The handle inside the toilet tank broke off, so I flushed the toilet manually, then stood watching while the tank filled back up, at one point getting distracted by my dog then the tank was overflowing pouring water onto the floor. I quickly shut the water to the toilet off and cleaned up the water, planning to contact someone to replace the very old innards of my toilet tank. Later in the day I went to my garage (directly below my bathroom) and there was water standing - like someone had dumped a bathtub full of water all at once. Nothing was dripping, etc. Then I noticed there was a small bit of water on the floor of my OTHER bathroom. What in the world is going on? It appears the water in the other bathroom is leaking from the water line that runs from the water valve to the back of the toilet - although this was not a problem before. Seems too weird to all be a coincidence. What in the world is going on?

Your toilet tank has a overflow tube installed to prevent the flooding that you describe. Are your toilet bowls backing up when you flush one or the other?

It appears the water in the other bathroom is leaking from the water line that runs from the water valve to the back of the toilet Are you saying , that in addition to all your other problems, that you have a pressure leak in one of the toilet supplies.

What really concerns me is this:

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The handle inside the toilet tank broke off, so I flushed the toilet manually, This is the simplest and easiest of toilet tank repairs and you let it slide. Why?
The first order of business is to replace and adjust all the tank parts. You supply leak should be fixed when the new ballcock's installed. If there's still a problem after that click on back and tell us about it. Good luck, Tom