2001 TXT still operates with key off

[TripleNad]

Just recently bought this cart. Came with light kit, but was disconnected. When I reconnected, I discovered the reason they were disconnected. Shorts inside the headlights, taillights not properly grounded, exposed wire all over the place. It was a mess. All lights and wires were replaced and is in working order now. What I did notice now though is that now the cart will operate with the key in both the off position and the on position. I took the plastic cover off the solenoid to see if there was anything that was obvious, and when I didn't see anything, I put it back on. At this point the cart wouldn't run at all. I took the cover back off, and the cart started working again. I checked the fuse and wire brushed a little corrosion off it, but it was fine. I pressed the pedal just enough to get it to idle, and then put the cover back on, and the engine died. After doing this several times, it finally stayed running and has continued to run since I screwed the cover back down.

I guess my question is: Where do I start? I have been reading a bit on here, but don't know where to start. Not sure if I can narrow it down to the solenoid, the key switch, or something else. Any help is apprecaited. Thanks!