Let me put it this way: She is having her cake and eating it too. She gets sex, validation and an ego boost from you, while she searches for the "man of her dreams" Probably some free meals etc too. Why are you allowing this? Do you like to be treated this way? I would NOT tell her about NC - what advanatge is there to telling her? by saying "I just need to go NC with you to get over you" it sounds weak. Just do it. Personally, I wouldn't give up the gym. So, you won't have a true NC....but NOW you are changing your life and schedule to work around hers.
I would just continue my life as normal, never initiating contact, rarely replying. And short, terse replies like you gave before. At the gym, I would be polite, but brief. If she says hi, reply Hi. But you DON'T have time to stand around chatting, etc. You are there to do your thing - do it. Treat her as an aquaintance. No more hanging around together, spending time together, long conversations. Just politely decline all invitations, saying you have plans. If she asks about the plans, just say with friends. It is no longer her business.
YOU ARE BEING USED FOR SEX, MEALS, VALIDATION AND AN EGO BOOST, WHEN IT IS CONVENIENT FOR HER, WHILE SHE LOOKS FOR ANOTHER MAN. GET IT?
She is no longer your girlfriend, stop treating her that way.