To keep the ground driven horse straighter, you need to run the long lines down thru the shaft loops if using a harness, along the horse sides. This puts the horse BETWEEN the lines, not under them, so you have much better total body control.
Whip is your aid, an extension of your hand or leg, with the driven horse. Whip should be long enough to reach the horse shoulder when you are behind him, both walking and riding in a vehicle. A whip that only reaches the rump is pretty useless to give help with. And I have heard it said that using whip on the rump will make horse "kicky" when they get irritated with you. I don't touch horse rump with my whip, so can't verify that information.
With the longer whip you can reach horse sides where your feet would hang, shoulder of horse, so a flick in those places can help horse bend, do some side stepping if needed to turn. I would be using a flick of whip on side, if he doesn't understand your vocal cue to start, to get him moving. We do horse name first, then command. "Holly, WALK on or Holly, TROT" Empahsis on the command and don't hesitate to touch horse with whip to get him moving when asked. Otherwise your are TRAINING horse to wait for second request and whip touch.
We most commonly use one-handed rein hold, in left hand with right hand helping to shorten rein for turns or shortening to stop. Whip is in right hand so whip can be used independently of the reins. Holding a rein in each hand makes any use of the whip interfere with reins and horse mouth. You lose contact on the right side when whip moves to touch-up the horse. One-handed rein holding goes by various names, but is not hard to learn, does give good control of the horse. And best, allows that use of whip without throwing away one side of the horse in motion.
There are a lot of nice LIGHT whips of length available. Those cheapy nylon covered whips are badly balanced, can be hard to carry for very long. Whip should actually be held all the time you are driving, ready to signal the horse, whack the dog running out to bite! Whip is no help when you need it NOW, stuck in the whip holder.