Read, read, read.
I've owned enough TWH crosses to know that your horse IS gaited.
Here is an interesting article to chew over: Direct Contact And Neck Reining With A Gaited Horse
An instructor is best, BUT, if that's not possible you can train yourself to direct rein your horse. Direct reining is STILL the way every horse gets broken in. It won't seem unnatural to your horse.
Here's a simplified primer:
WALK--horse bobs his head, every other step. Count the steps and follow the forward and backward movements. It may help if you hold your reins as if you are driving the horse.
TROT/RUNNING WALK/AMBLE--horse maintains the same head position. Post the trot (if he trots), rest your hands with light contact in front of the pommel.
CANTER--Like at the walk, horse puts head forward to "grab" when the back feet hit the ground, then moves head backwards, completing this motion when the leading foot hits the ground. Practice feeling the motion on a loose rein, then practice following it.
It isn't rocket science. Somebody videoing you will help, too.
One more thing, get your horse desensitized to your legs constantly resting at the sides. MANY western riders try not to touch their horses sides so that their horses remain super light to cues. English-ridden horses can be equally light, but learn to ignore the legs when there is no cue.
Hope this helps!
Here is a picture of our horse, "Tyke" (1970-1998, RIP) He was a QH/TWH cross and we did practically every discipline with him and rode him all over the place at home and on vacations.