it's a 10 year old craftsman mower 18 hp OHV.

Below the carb is a bowl with a plastic float. Screwed into the bottom of that there appears to be a jet that has power to it. I think it is actually a plug, when activated it stops gas flow into the bowl. maybe an electronic gas shut off thingy?

When the mower is turned off this device got hot to the touch, indicating to me that it has power, and it's probably not supposed to. I wonder if this piece could be bad?

I need to install a shutoff to keep this from happening again clearly.

The solenoid of which you speak basically blocks a circuit in the carb to lessen backfire/ emissions when shut off. If you feel a tic or hear a click when the the key is turned to run- it is working. You can remove it and watch it actuate too. It will not however prevent the condition you are having.
You may have a touch of debris between the float valve and seat which is allowing the fuel through. If there is a worn "ring" pattern in the valve- replace it for sure or do it anyway. Clean the tank and bowl, flush the lines and thru the float valve w/ fresh fuel, replace the fuel filter, change the oil (plus oil filter if equipped). Hook the spark plug wire over the torx headed bolt and crank the engine to expel more of the fuel.Reinstall the spark plug and wire and expect more emissions for the first few minutes. remember that the muffler may have oil, fuel washed down inside of it and tho possibility of fire is slight, prepare for it none the less.
The condition may still reoccur. An inline shut off valve kit is a great thing to install and use. That way, if a failure happen whilst it is sitting- the engine and muffler won't flood.

I think I'll be checking the operation of this solenoid when I get home....

So either debris kept the float open or the float valve isn't seating all the way but i just read in another thread a leaking needle and seat shouldn't cause this big of an issue.

I think all the debris came from the insides of the dry rotted gas lines i just replaced.

Got a shutoff valve to install tonight, i read it is suggested i just turn it off and let the motor run out of gas.

should i install it upstream or downstream from the gas filter?