Are there any enemies around? Once a unit enters combat mode after spotting an enemy, their behaviour turns entirely erratic. E.g. civilians might freeze entirely and just crouch in fear, or they might start running in any odd direction and screaming. Soldiers go into slow-mo mode and start moving using bounding overwatch. And vehicles, from helicopters to trucks, go off road and generally start acting crazy. There is, sadly, little to nothing you can do about that
That said, vehicles in Safe mode SHOULD stick to the road in almost all situations. It's very odd if they don't do that, but instead as you say start finding shortcuts. I mean - they'll swerve and run into trees and streetsigns and be unable to turn a corner without destroying a few metres of someone's fence, but out in the countryside they're usually pretty good.
If you can't get it work, try posting your mission here and we can take a look at it
Wolfrug out.