You go to a store and apply at one of the job kiosks. You can put the application into any Publix and apply to every Publix in your metropolitan area. They are a promote-from-within company, so expect to start from the bottom. If you learned how to cut meat with your experience as a meat manager, then you can get hired as a meat cutter, which is actually the highest paid hourly job in the company. Otherwise, you'll have to start off as a meat clerk and get promoted to meat cutter. After that you can get promoted to assistant meat department manager, then meat department manager. After that you can get promoted to retail improvement specialist. RIS's are the guys who basically go from store to store and tell all the meat managers in the district what they're doing wrong. If you don't listen to them, then expect to feel the divine wrath of the district or regional manager when they inspect you--or after you get a low Steritech score.
You can also go up to assis. store manager(though it's usally the customer service manager who gets this job), then store manager, then district manager, then regional manager, and some regional managers have become CEOs. That said, the odds of you getting promoted to RIS or Assis. Store Manager are extremely low. I know a bakery manager with 27 years of experience at the company who never got promoted to anything higher. Still, the meat manager makes between $60000-$80000 a year. Although they tell you not to work past 53 hours a week, you could end up working 80 hour weeks on holidays. While you do have to start at the bottom(they virtually never higher people as managers), I've seen people get promoted to department manager within 2 years. Management experience greatly helps(which you have), as will have any college degree whatsoever(even if it's a philosophy degree or an associate's in culinary arts).