I'm debating with this lady and after debating a bit and committing tons of logical fallacies, I point them out to her, and she responds saying, "I don't care about committing logical fallacies with someone like you."
If somebody starts pointing out all the (alleged) logical errors in somebody else's arguments, it's probably going to be perceived by that second person as an expression of superiority. People don't like it when other people try to claim rank on them and they instinctively resist it (whether effectively or not).
Criticizing somebody else's logic might even be an ad-hominem fallacy in its own right, if it distracts attention away from the actual argument that a person is making by putting them in the position of having to defend themselves personally.
Rhetorically speaking, it's probably more effective to leave the other person out of it and to direct your comments to the other person's argument. If that argument is logically consistent with potentially devastating counter-examples (and that's often the defect with the fallacies), then gently suggest an effective counter-example. If the defect is one of relevance, then inquire as to how the other person's point supports his/her conclusion.
It's often possible to do that without pushing the other person into a position where their ego-defenses rise. Sometimes it helps to affect a little false-modesty, by suggesting that you might not fully understand the other person's argument, and by asking them to expand on it. That inflates the other person's ego by giving them an opportunity to play instructor with you. Then drop in your own counter-points in a friendly conversational way, pointing out what you believe the problems are in the argument the other person is crafting.
It helps to express some common purpose with the other person, reassuring them that you share some of their fundamental concerns and goals. Then gently point out that you simply don't understand how the other person's argument can succeed as presently conceived and expressed. Suggest that it might need some more work.
That puts the two of you in the position of being allies, both working towards shared goals and both struggling to discover the best way forward. That's going to make the other person a lot more receptive to listening to your suggestions.
That's the bottom-line, I guess. Never put your opponent into a rhetorical situation where he or she perceives agreeing with you as a humiliation. That will pretty much insure that they won't agree. Always try to give the other person a path to agreeing with you that leaves their pride intact.
