Not allowing you to get down her offside is an evasion. If a horse finds it difficult or uncomfortable to run a circle in a particular direction, they may use several forms of evasion to avoid it. If one of these avoidance tactics doesn't work, they will try something else, and this will result in the horse facing the handler and the handler not being able to get into a position to send the horse on.
Horses which have learned such serious evasions have to be taught to yield their head, neck, and shoulder, and forequarters, initially to a training stick and, later, to the handler stepping in toward those areas. Then the horse can be caused to yield its fore end away from the handler, and then sent on on a circle around the handler. Any potential for the horse to fall in and attempt evasion can be counteracted by the handler stepping in toward its shoulder and then sending it on again.
The above it correct. you have to learn how to first send her head and shoulders away from you, so that she is no longer facing you square on . ONce she does this, you progress to having her move forward.
When you went under her neck and tried to put your hand on her right side neck, and she "went backwards", this is when I would go with her. I would just keep "asking " to be on her right side, and not allow her to back away from my request. So, you move with her but stay awar of the milisecond when she stops backing, at which point you ease off a tad. you should be on her right side, walking with her, this whole time. So, her backing away does not remove you from her side, but her ceasing to back away will earn some relief.
IN general, when asking her to turn her head and shoulders away from you you, don't let her attempt to avoid you by backing make you stop asking. you can hold the rope and give a few yanks on it to indicate to her to go forward. A steady pull is better, but if she is backing , you can break her out of that with a few yanks, then a steady rope contact to say "dont' go backward" and a pressure on her head and shoulders (with whip)to make her step away from directly facing you to the place where her right side is toward you. Reward that and ease off .
Eventually, you get that step out and then you ask with the rope and pressure more on her hip (with the lunge whip or long dressage whip) to have her move forward.