Download the manual as Mark indicated. Look on page 79. Your starter solenoid is part of the starter assembly(as opposed to separate from the starter like it is for the 1000 series). You need to determine if solenoid is bad and the contacts are sticking closed, or something is still causing the solenoid to remain on even with the key off.
First, here is how it works 'normally'. Look at figure 39.10 on page 79. The smaller gauge blue wire is what provides the 12 volts needed to engage the solenoid and start the tractor. That blue wire should only be energized when all safety switches are in their proper position, and the ignition is turned to the start position. When you put the key back to any other position, the voltage goes away. You need to pull that blue wire off, and place a meter or test light between it and ground. Proceed to try and start the unit, and watch the light or meter as you bump the key to the start position and back. Does it reliably go on and off? If it does, then it is highly likely the solenoid contacts are 'sticking' closed and the solenoid will need to be disassembled and the contacts burnished, or just replace it altogether.
BTW, it's not unusual for the red wire to get hot if the starter is running for more than the typical time it takes to start the engine. There is a lot of current flowing for a longer than usual time in your situation.
Another way to test this is to try and start it again, knowing that it is likely to continue to run. Take a pair of needle nose pliers and while the starter is still turning, quickly pull off the blue wire from the solenoid. Be prepared to pull the battery cable again. If the starter is still turning with the blue wire removed, it's the solenoid that's the problem. If the starter stops when you pull the blue wire, then you have a different problem. Let's for now just determine if it's the solenoid or not.
Let us know how you make out!
