Is your horse tied up in her stall while you groom and tack up? I'm assuming yes, I only ask because I occasionally just groom mine while he eats his hay. Having her tied may increase your confidence, if she isn't. Just be careful grooming in such an enclosed space, and try to never let her get between you and the door (not to make you more nervous, honest, just another safety tip)
I find that keeping both hands on the horse is helpful. Brush with one hand, and pet or touch with the other. I'm not sure why it helps, but it does.
Lesson horses do have a tendency to be a tad nippy when they're cinched, just a side effect of so many inexperienced riders jerking the cinch tight. I would put my elbow in her way while I gently cinch her up. Don't chase her down and "get her," just let her nose run into your strategically placed elbow when she nips in your direction. You probably won't cure her problem, especially if she's a lesson horse in my understanding of the term, but she might learn that YOU won't accept the nip threats, and that YOU will be gentle. With the little kick, she may be a bit ticklish and you found the "not so sweet spot."
Just stay as close as possible when you groom. Most horses find it comforting, and it's safest for you. If you're still clean when the horse is, you didn't stand close enough
. You will gain confidence with time, 6 or 7 lessons and groomings isn't that much. You've gotten a lot of good advice, just give it time.
Good luck and have fun!