As you ride together more two things should happen:
1. You will learn how to compensate for her wandering before it even happens, and
2. You will learn to correct her without even knowing you're doing it.
Notice, please, that neither item has anything at all to do with changing your horse's basic behavior. As Racer said, it's YOUR responsibility.
Your horse should, however, respond and stay on trail with just the tiniest, almost automatic, aids.
But, when all is said and done, the goal is to get out and ride. And ride. And enjoy.
BTW, if you don't mind my asking, where do you live where there's chest high snow? I can think of lots of places like that (some very close to home, actually), but I'm curious anyway. Tnx!
"Don't just feed it! Ride it!" And the People said "AMEN!"