You are correct, running-on is called dieseling cause it is using the heat to combust the fuel like a diesel does without the need of a spark plug. There are a few things that can cause this but most likely it is too low of octane gas. I would assume you should be running at least 92. Are you running 89 or even 87 octane?
Another cause is too high compression ratio. Other than factory, the reason the compression ratio may rise could be anything in the combustion chamber taking up space such as too rich a fuel mixture, carbon buildup, etc.
Another possible cause could be too far advanced ignition timing but that would most likely causing pinging (pre-ignition) as well.
Actually too hot of a spark plug might cause it too. Anything that would cause either the compression ratio to be too high, the fuel to explode pre-maturely or anything that may cause a flame front other than what is normal (spark plug).
If it is carbureted the throttle not completely closing could cause it, hence the need for an anti-dieseling solenoid. If fuel injected, leaking injectors.
First thing I would do is ask what octane gas they usually use. If they would allow you to and if the tank is less than 1/2 tank add to it whatever would bring the octane up at at least 92, so if it has 1/2 tank of 89 add 1/2 tank of 94 if you can find 94, to bring it up to 91 1/2, close enough.
If it running too rich, most likely it will be blowing out black smoke also. Too much carbon and it may blow out blue smoke.
Ask the owner if you can pull the spark plugs and have a look at them. I don't think it's nothing major, simply having the ignition timing off a little and the idle set too high would cause a run-on.
Sorry about that I was up since 4:30am, I fell back asleep.
I'd say at about $2 each for spark plugs, $16. Probably about 1 hour to install them, set the timing and adjust the idle or about $75. Total $91.