All very good suggestions so far.
I'll try to keep from rambeling but for some reason I like this thread, so forgive me
One of the ideas, a heat source causing the damage is possible. As soon as you noticed the problem did you by chance feel the cable in heat exposed areas to see if the cable was hot to touch?
As far as the cable being used as a "ground" that is also very possible. IN order for current to flow there must be a circuit. Both ends of the cable would have to bond with something that is able to complete the circuit to ground. The first one is obvious and is at the engine hook up. The next location would have to be the throttle lever device.
I think this is why some control cables have teflon or plastic type insulators at one or both ends to prevent current from flowing through the cable.
I looked at this part
diagram on the colemanequip site and I am unable to determine if such insulation is provided at the throttle lever. Are you able to inspect this linkage to see if you have any worn or missing devises intended to insulate against current flow?
If there isn't, than quite possibly you could make your own. Go to the auto parts store and look for small enough teflon type washers to place inbetween the clevis and linkage arms. If not able to do that, how about a pice of electric tape to the top and bottom on the inside of one end of the clevis. Of course now you would have to replace the pin that secures the clevis with a non metallic type. If you are able to comply with this than you would break the circuit and current can not flow through the cable.
I would think a non mettalic tefflon pin would be ok since the friction you feel in the accelerator lever is a "load feel" set at the lever and the cable linkage is not actually having to transmit the same force.
For curriosity sake, you can see if and just about how much current the cable is carrying. First we can see if the cable is bonded with the tractor. Disconnect the clevis from the engine and using a meter set to ohms touch one of the leads to the clevis and the other to a good ground. How much resistance do you have, if any. This will let us know if the throttle linkage is bonded with the tractor. If it is then it is going to carry a certain amount of current when hooked up to the engine side.
By default if the cable is indeed bonded from the engine and completing a circuit with the throttle lever using a meter to check for voltage isn't really gonna do much good. No matter how small the amperage the meter will always say 12v dc.
One way to see just how much is going through there is to get a test light. Any light will do but something like a tractor add on light would be best. At least something in the neighbor hood of 50 watts. This will equal ablut 4 amps. It will most likely take a lot more than that to weld the inner parts of your cable.
Set the light up safely with long enough jumpers. Connect one end to the positive terminal of the battery and the other end to a clevis on the cable. The best location would be the clevis on the throttle control linkage. Be careful with this and provide good safety measures for yourself by maybe even splicing in a fuse for the test light and a switch. It's easy and most people might even have extras of this stuff in there garage.
With this hooked up, turn on the ignition but not the engine yet to see if it illuminates. Whit a helper, start the engine and see what the light does while cranking. What doe it do after the engine has finished cranking and is running?
One other possibilty with a new cable is to leave it disconnected from the engine side. Wrap the clevis and any metal on the cable with tape to insulate it and safely secure it. Now the tricky part. Fabricate a way to either control the engine side with a stiff cable to the driver or safely secure the engine linkage so the engine doesn't go whide open uncontrolable on you. Now go out and drive the tractor for a while and bring some adult beverrages with you.
When you are done, inspect the control cable to see if it is damaged like the others. If it is than this is not from current flowing through it. This would have to be heat damage.
Sorry that ws so long and I hope it ws of some help.