When she is pushing your hip and knocking you off balance, she is showing dominance, and telling you to get out of way.
By letting her follow you off lead, you are getting into a nasty habit of letting her make the decisions. Never a good thing.
Each time you have walked along with her loose, it is making her think she is your leader, this is evidenced by the fact that she is walking off when you are grooming, or being pushy when you attempt to lead her.
Quit doing anything with this mare where she is not haltered and under your control. Each time you don't have control of her, and she decides what she is going to do? She is getting it more ingrained in her head that you are her inferior.
It will trickle over into every other aspect of owning/riding her eventually.
Do not let her get close enough to push against you, that is a bad thing right there. Distance generally should be elbow length or little more, and if horse enters closer, correction with lead rope to move horse out of space needs to be done immediately, and not with yelling/flailing away, but a quiet and calm, "back up" or even just correction with lead to put her where she needs to be.
Also, make sure you are not giving mixed signals. By this I mean one time she can follow close? Next time you make her stay back. She will have no idea which time is which, so better to set ground rules now that will benefit both of you.
Horses make me a better person.