If you just go in and plead guilty, then yes it will be on your record permanently. Lucky for you, you do not have to do that. Talk to a local attorney to see if the court where you are charged has a diversion program. Diversion is set up to have you do a variety of things that are designed to help you not make that dumb decision again and, if you do them, the case will be dismissed.
You need to hire an attorney to inquire about a pre trial diversion program that could keep the charge off your record. Because you were not arrested, you are in a good position to keep your record clean. Do not just plead guilty just to get it over with because it will be on your record permanently. The ban from Publix may be negotiated but is usually a component in some form or another.
DISCLAIMER: This answer does not constitute legal advice and no attorney client relationsip has been, or will be, created until a valid engagement agreement is signed. No duty arises from this posting. Answers posted here are general and made with limited knowledge of the actual facts of your case. Always speak with an attorney licensed to practice law in your jurisdiction if you wish legal advice specific to your case.
Paying the fine is the same as pleading guilty and will result in a permanent criminal record.
Hiring a lawyer should be your next step.