Hey Stella! I never thought about acne as being like whack-a-mole lol, but that's a pretty spot-on analogy. So, you've tried new toothpaste... how long have you been using it? Also, do you do anything else to irritate your chin (i.e., I personally and naturally prop my chin up with my hand or fist sometimes, which pisses off my skin). Some people (WishClean and Wombat666 at least) have said that xylitol may be an issue. It looks like Nature's Gate toothpaste has xylitol in it, so maybe that's something else to consider.
The be-all, end-all solution would be to just stop eating / fast for a while so you don't have to brush your teeth... (kidding of course... keeping narrowing stuff down!)