Starting problem

Unplug the small red wire on the solenoid. The solenoid should turn off. If it does, check for voltage on that red wire. It should be 0 volts unless a helper turns the ignition switch to "crank". If there's always voltage there, check that wire at the ignition switch. The switch would have to be sticking, out-of-adjustment, or the wrong one.

Do you have the round solenoid like in this photo or the older square style? If it's the square one, does it have one or two small terminals on top?

If you're finding 12 volts on the smaller terminal when the red wire is unplugged, the solenoid should be engaging. Double-check where your voltmeter or test light is grounded. If it's grounded incorrectly on the positive battery terminal, it's going to look like there's voltage on the solenoid terminal.

I assume it's not cranking with the red wire unplugged. When it won't stop cranking, unplug that wire and see if it stops. If it doesn't, tap on it. If that makes it disengage, it's sticking internally and must be replaced. Remaining engaged too long will overheat it and cause the coils of wire to warp. If it still doesn't disengage, suspect the two large cables are reversed.

If the two large cables are reversed on the two large studs, the solenoid will remain off until it is engaged, then it will stay engaged. That's because the solenoid coil is actually two coils of wire. One is grounded to the metal mounting bracket, but the other one is attached to the starter cable's terminal and gets grounded through the starter motor. Once the starter is energized, that coil is turned off by having voltage on both sides of it.

With the two large cables reversed, battery voltage is applied to the stud that has the one solenoid coil attached to it. THAT would cause you to measure voltage on the small stud when the wire is unplugged. That puts the two solenoid coils in series so they'll draw much less current than normal but there will be a constant drain on the battery, ... Enough to kill the battery overnight.