If you can post some videos of your riding this horse on the flat and maybe over a ‘course’ of poles ridden like a 2’ course, you might get some valuable feedback. The signs you’ve described: speeding up over jumps, refusing jumps, swishing tail and pinning ears, are all clear indications of discomfort, whether that’s ulcers/back/saddle fit, or lack of confidence from you frequently cantering to a big miss and catching him in the mouth or back. Without video, we can only assume the worst from your description.
Have you experimented with different tack at all? Maybe a softer bit, borrow a friend’s saddle, etc. Go back to flat work, work on suppling both of you. If he was a decent dressage horse, he should understand spirals, lateral, changes within the gaits. Schooling all these will help you trust eachother a little more. Don’t work canter until you can do all of the above at the trot. Don’t try to jump until you can do all of those at the canter. Then, if things have improved (less ear pinning, light in the bridle and to all your aids) you can think about the next step. The next step is NOT jumping. Next you introduce poles. Lots and lots of poles! Trot poles until you’re both bored. Then canter poles until you’re both bored. Do ‘courses’ of them, practice getting 1-2 more strides in a ‘line’, then 1-2 less, until you can switch back and forth perfectly. Develop your ‘eye’ and your horse’s confidence. Then start making a few of those poles into low X’s. The second the horse starts to speed up unasked, or refuses, you need to back up two steps.
Think of jumping a course as a house, and all the flat work detailed above is the foundation. If you try to put a house up with no foundation, the house will crash and burn, just as your current jumping is. You need to build a strong foundation, brick by brick, to have a strong house. The bigger the house (jumps) the stronger that foundation needs to be. There’s no timeline associated with this. You can’t be in a hurry.
It sounds like perhaps this wasn’t the right horse to purchase for a rider with jumping aspirations but without deep pockets to fund training rides. However, if you have a vet (and possibly also a chiro or bodywork specialist) give the horse a good going over, and all looks okay, going back to the very beginning, as above, you may be able to progress to jumping small courses successfully with this horse. However, once a horse has ‘learned’ to refuse, they never unlearn it, and it may always be a problem. Perhaps his true calling is as a LL dressage horse.
One more note. Without pictures it’s hard to know. But many paint horses are built downhill and naturally on their forehands. A horse with this conformation will always find jumping harder than a horse with sporthorse conformation. That may be another complicating factor.
One more note: based on the description of the escalating jumping problems, and the fact the horse had very little experience as a jumping horse, which is what you purchased him for, I’d be asking myself some hard questions about my trainer. The fact that he/she encouraged you to buy this horse, isn’t coaching you on how to solve the problems, and hasn’t jumped the horse him/herself to try to figure out the issues, means they may not be the right trainer for you. It might be time to take a good hard look at the program, the quality of instruction, and the progression of your riding and that of your barn-mates. Just my 2cents.