Flouride toothpaste, topical steriods and other creams are thought to possibly contribute to something called "perioral dermatitis" which looks like acne and affects areas from the nose to the sides of the lips and down on the chin. Hormonal acne is more likely what you have and affects the lower part of the face as well, but not as noticeably circling the mouth. Hormonal acne is not caused by toothpastes or creams, but by the hormonal balance in your body. Of course you are right- many things will aggravate acne. If flouride seems to be one of them, you could try cutting it out.
But I doubt removing certain toothpastes will make a difference (remember, a lot of water has flouride in it too). Even doctors trying that for their perioral patients have seen few conclusively positive results. So you could go ahead and try it, but my advise is to get treatment for hormonal acne, because that's most likely what you've got.