One thing you can try (after you have more confidence and better seat from some lessons) is when he starts to get too fast, work him into a circle. You pick up one rein and start circling him . He will have to slow down. When he does, immediately let him out. He will speed up. Circle, slow him, let him out. You will have to do this a thousand times, but eventually, when he starts to speed up, just him feeling you pick up ONE rein, will tell him that if he doesn't slow down he is gonna have to do another Gol Durned circle and he'd rather not. So he will slow with just that. And you must immediately drop that rein to reward his slowing. He'll start to slow from just picking up one rein.
However, this will not change the fact that he will want to move out rapidly most of the time. TWH just are that way. Once you are more comfortable with his gait/walk, you will stay calmer on him and then my bet is that he will not panic speed into a canter. Horses run away from fear and pain.
So, if his mouth is sore from the bit, he will only fun harder against the pain when you apply the bit to slow him down. Gotta let his mouth heal and then figure out why the bit is rubbing him raw there.
Applying the bit strongly isn't a crime. After all, the horse will have light pressure once he learns to come off it. The trick is that you have to remember that you want him to come OFF that pressure, not ride it.
So you have to be as strong as it takes for him to give to your pressure, which is just a bit more than equal to his pull on the rein, then when he gives to you, you have to give the reins, totally. He will learn that there is relief for him the second he gives to the pressure and there is NO tug of war. However, it can look ugly for a few minutes when he pushes against the bit. you have to meet his pressure and an ounce MORE, so that he does give, then it's total freedom for him until he tests again. So, one may have to be really strong for a short period of time to make a horse soft to the bit, especially if they have learned to lean on the bit and go right through it. I would go back to a snaffle for awhile and maybe get a Dring so that there is no pinch risk at the corners of the mouth.