ok I've never really had this problem with a cart horse because most of mine learned to ride first and since I ride reining horses, whoa is one of our most important maneuvers to learn. You simply can't slide without a GREAT whoa. We really start with this from day one with weanlings and yearlings. We do it on the lead then under saddle so that whoa is always respected. I only use whoa when I know I can get a stop. Otherwise I use 'easy' to just slow a horse down.
I'm having to do it a tiny bit differently with the mini's but it's still the same principle for me, I'm just doing it all in a ground driving situation with some modifications.
But here are some things I'd do if whoa was a big issue. Some might disagree and I'm not saying you have to do it this way. But it would be what I'd do.
Whoa is taught as a back. I want my horses immediately thinking back when I say whoa. Not only does it help them learn whoa, it teaches them to gather their body in the stop. With my reiners, we do it with three different cues and we test them independently. I want a horse stopping off the rein, off the voice and off my seat, independently and together or in combination, sometimes just voice and seat, sometime just reins and voice etc. But I proof that whoa to where it's there for all of them and in all 3 gaits. Obviously you'll eliminate the seat, but I still want the other two to be responded to. The horse needs to come back to just your hand and he needs to stop with just your voice, independently and together.
So when I do ask for the whoa I always back a step or two sometimes more depending on how quickly the horse responds. General rule of thumb is for every step they take over my stopping point, they have to back to that stopping point. The quicker they learn to back with a whoa the quicker they'll learn to down shift those gears and eventually they start to relax into the stop and you won't need to back. Most horses don't really WANT to back at first and a tense horse will have a hard time doing so. Once you get them thinking back anytime you ask for a stop, whoa is hardly ever a problem again because it gets them to think first and in turn to relax. The body position for stopping is the same as for backing, so they need to learn to get into that position. The transition into just stopping without any steps back isn't that hard to achieve and one tiny step back never hurts anything. My reiners will stop so fast that they often have a foot in the air that they just change directions mid stride. It's really kinda cool when you get that, because then you know your horse is ready to move up to a faster gait and eventually slide. Of course you won't really need this kind of precision in the back but I don't find it hurts. You can always 'lazy up' the stop so that it's slower and gentler. They learn that much quicker than they learn to shut it down completely
Also, I practice my whoa when they're good and tired and ready to stop. I don't bother trying to get it at the beginning of a lesson if they're amped up. With the riding horses they always start the day with a back. I've started doing this with my minis when I'm ground driving as well. We don't just RUSH to take off. They learn to stand to start because we're not going anywhere until they're relaxed and the first thing we're going to do is back up. Then once I start moving, I don't really bother with whoa until they're good and tired. The whoa becomes a reward, a rest from the work we were doing. My horses learn to enjoy stopping and resting
They nearly always drop their heads and just relax, they know if they don't take the opportunity to relax, they'll be in for more work. I also practice transitions, over and over and over. W-T-L, T-W, T-L, L-T, L-W, W-L, W-T etc, All of them, over and over. so that they're comfortable with them at any stage. I find the downward transitions easier than the upward transitions but they need work on all of them in addition to the whoa, it helps establish the downward to the walk, and eventual stop as a ritual thing for a cart horse. When I go places if the horses are amped up our warm ups consists of transitions, over and over.
Also, I don't know if you're doing this or not but it's a common mistake so I'll mention it, but make sure you're not hanging on the horse's face when they do stop. I see this with alot of newbie riders and even some very good riders, they think that more pull equals a better stop and they think they need to hold the horse in place. It's not the pull it's the release. Reins that are too tight or jerked too quickly will make a horse jig especially with a sensitive responsive horse. They feel the tension and they get nervous. Soft slow tension on the reins with a quick release as a reward for standing still will achieve more than if you haul back on the reins with all your might and expect them to shut it down. It's not like a car's braking system where you have to apply pressure and then hold them there. Soft hands with a good feel rule! Even horses who are on the bit need a release for stopping nicely. You want your release to be much quicker than your cue at least 3x different. Pull back slowly, count it out one-two-three don't jerk then release quickly as soon as his feet have stopped. Allow him to stretch his neck and relax then find the contact again when you pick up to take off again.