If there is burned wiring then I'll agree that there probably would be issues there but that's something that would be impossible to tell without going into the computer and looking for, and troubleshooting any codes found. The only thing about that is if it were an electrical issue caused by a burned/shorted/open wire then the problem would more than likely be a constant thing and not semi intermittent, brought out by using one function, etc, etc.
The same holds true for a bad solenoid as a burned solenoid will cause constant problems vs intermittent problems.
That said, the last machine I ran into that had a problem similar to what your saying is happening was a CAT with a bad main pump. What happened there was the surface the swash plate rides on had become worn and uneven. In normal practice that surface is polished smooth so there is no binding, etc as the swash plate changes position and therefore the output/flow from the pump. When digging repatively the flow coming from the pump stays within a relatively small window so the swach plate isn't moving all that much. When you start to travel the flow required by the track motors causes the pumps to go to nearly their full flow, and full pressure to operate them. As far as the flow is concerned this means the swash plate is making a large move to cause that increase in flow, and on most excavator brands the system relief setting when in travel mode is also increased above the normal setting for other operations. In other words when you travel the pumps are working their butts off to keep up. Because of this problems with the pumps will typically show up when traveling even when you see no problems with any of the other functions.
Assuming that what I just talked about is what your seeing, here's why. Like I said earlier the surface the swash plate rides/pivots on has to be smooth to allow a seamless transision from one flow rate to another. This is happing constantly to some degree simply because the flow rate is directly tied to the pressure when it comes to these pumps. In other words as pressure rises, flow decreases. This happens in order to keep the pump pressure and flow output within the power limitations of the engine. When the swash plate can't move freely and is binding, it can cause the machine to do all kinds of strange things just like your describing. In the case of the other machine I mentioned even the part about letting it cool down and the problem seeming to go away for awhile was the same. Typically what happens there is a combination of the cold oil leaving a heavier film between the mating surfaces, coupled with the increased tollerances between the cold parts allows the swash plate to move a bit easier than it does when the oil gets hot and the film seperating parts shrinks and also the hot parts get larger and the tollerances shrink to a minimum. Inside those pumps tollerances are often down to thenths of thousands so it doesn't take much at all to cause one to start acting screwy.
Unfortunately the only way to check for a problem like that is to use a flow meter with loading capabilities. I know this probably isn't what you wanted to hear but, unfortunately, it's the the only way to check for problems with that type of pump with any degree of certaintly.
One last thing. An excavator is designed to work, not just sit around looking pretty. In other words what the guy you rented it to was doing was exactly what these machines are designed to do. So, unless he ran it low on oil, beat it up, or otherwise intentionally did something to screw it up, what happened more likely than not was not his fault. He was just unlucky enough to be the one operating it when the problem decided to rear it's ugly head. In other words, what I see when something like this happens, is that the problem was already there and it was nothing the guy operating it did, or could have done, and it was nothing you did or could have done to prevent the problem from happening. So, don't beat yourself up thinking you made a mistake renting it out and trying to make some money because with something like this it all comes down to the old saying, "$hit happens".
Good luck and as you get into the problem further and have any questions feel free to ask. My email is open if you want to do it off the boards.