Well...if the girls are saying that you didn't offend them, then I don't see any reason for you to believe you did. Maybe you're right, the assistant manager does have a problem with you. I had a similar problem with a manager a few years ago...she was a pathological liar. She had caused trouble between me and new employees making it look like I didn't like them, which couldn't be further from the truth. I found out later she had exaggerated it. because of that, I would never work in her new salon she opened.
I don't think you did anything wrong talking to clients, especially if they asked you for help. After all, I really believe that we can't OWN a client! I also don't blame you if you want to talk to the DM. If your salon is not upholding the company policy, then you need to hold a meeting, all of you to see that they do in the future.
I think you should confront the assistant manager too. Tell her that this is upseting you. And I think it would be a good idea to have a shop meeting with her, the staff, and the DM as well. Keep everything in the open. Good luck, keep us posted!