There are two things that control whether your starter motor is getting power: the key switch and the starter solenoid. I'll write this at a pretty basic level, just in case you aren't familiar with how they work.
The key switch sends low power 12 volts to the solenoid. That wire is connected to the windings inside the solenoid that cause the heavy-duty contacts to close, sending power through the large cable to the starter. Either your key switch is always sending power to the solenoid, or the heavy duty contacts are stuck in the closed position. A third possibility is that the winch control got wired in wrong and it is sending power to the solenoid, but I think you would have noticed that while installing the winch. That said, when something goes wrong it is always a good idea to think about what was worked on most recently and if it could be causing the problem.
I don't have a Trail, but the starter solenoid can be found by tracing the cable from the starter back to it. On an X, the solenoid is a few inches forward of the battery. I suspect it would be in a similar spot on a Trail. Find the solenoid and disconnect one or both of the skinny wires. If the starter stops running, the problem is probably in your key switch. It is possible there could be a short in the wiring harness, but it would have to be between a "hot" wire and the solenoid control wire and that is far less likely as a bad switch. (This is also the right time to think about how the winch control is wired.) If the starter continues to run with both skinny wires disconnected, the contacts inside are stuck. You might get it to release by tapping it with a hammer, but that would most likely only be a temporary fix.
If you need to replace the starter solenoid and don't think you can get an AC solenoid before your vacation, older Fords used an external starter solenoid that is electrically the same as your AC solenoid. Your local FLAPS (friendly local auto parts store) will likely have the Ford part in stock, or available within 24 hours.