Champion fan relay, odd shutoff issue.

the thermostat switch is just a switch to ground.(IIRC)
You could bypass it with a regular switch you controle from under the hood. Nothing wrong with letting the fans run even while you are driving.

Not sure how you wired them but I hope the main power for the relay for the fans is comming directly off the battery, rather than the cars wiring

Something to remember is if the engine is hot but just shut off the water temp tends to rise, when you turn the engine on cooler rad water starts to circulate and that might be whats turning off your fans.

If you sit in a parking lot with the engine running after a stretch on the interstate, do the fans ever turn on?