The Walgreens at 5501 S Kirkman Rd (Kirkman & Vineland) is probably the closest if you're staying at Hard Rock or Portofino and they do have a full liquuor department according to the Walgreens website. You could even walk to this location from Hard Rock and Portofino (it is about a 1 mile walk), though I'd recommend the cab for safety.
While prices may be a bit higher than Publix, if you're just looking for beverages (adult & otherwise) and snacks, you'll do fine here. You can sign up for their Balance Rewards program to save money.
In addition, there are plenty of casual restaurants here where you can get lunch - Alehouse is almost right next door and they have daily lunch specials that are going to be better than any fast food place, plus drink specials and a good selection of draft beers if you are a beer drinker. There is also a TGI Fridays at 5901 S Kirkman (just down Kirkman about 1/2 a mile) which has inexpensive HH specials on appetizers and drinks and an IHOP at 5203 S Kirkman (1/2 mile down Kirkman in the opposite direction) if you're craving breakfast foods.