If she's allowed to have done this for years for the previous owner, it IS going to take lots of TIME to fix the issue. And it may never go away completely, and you may always have to "keep an eye on it".
As far as her issues with the bit, has she ever had her teeth worked on by an equine dentist? She may have a mouth issue.
Since you say she doesn't turn well in a bosal, I would treat this horse as if they were never broke to ride. Go back to square one.
That means teaching her good ground manners when being led. That means teaching her to give to pressure in a halter while leading and being handled. Teaching her to stand still when you saddle her, bridle her, and mount. Teaching
her to give to the bit or bosal (from the ground first), because if she doesn't turn well, she needs re-training.
For the trotting issue, serpentines work wonders. That way you aren't hanging on her mouth with both hands the whole time, making her ignore the bit. Use one rein at a time to serpentine her back and forth. I do this with my horse who is VERY HOT and can go all day. You've just got to be able to last longer.
When she walks nicely, leave her alone. When she picks up a trot without you asking, serpentine to slow her down. This will be a long process to break her of, if her previous owners let her do whatever she wanted.