NMB is nice this time of year, less tourist than MB. Only bad time of the year to be in NMB is Memorial day weekend. All the rice rockets hit the town, traffic is bumper to bumper. As far as fishing, you can either hit the piers in MB or make the trip to Cherry Grove.
For the gunshops, We have two good Ranges. The one i work at has been damaged by fire, not sure if it will be up and running by the time you get here. Myrtle Beach Indoor Shooting Range. The other is down on the south end, in Murrells inlet. Some people complain about those guys attitudes at Coastal Sports, i've only been there once or twice since they won't allow you to fire more than 3rds from your AR, and thats only if your sighting in a scope.
ETA, you'll only need a license to Saltwater fish if you are on a personal watercraft. AFAIK, no license is needed to fish from the pier, or surf fishing.