There is a great Mexican Restaurant right there 50 yards from the hotel, Los Tres Magueyes. The Tres Magueyes dinner with shrimp, beef and chicken is outstanding.
Apex Wings is 3/4 of a mile up the road your on at the hotel, 55/Williams Street towards town, but that's where we are eating Saturday late afternoon/evening.
My recommendation would be to get on US 1 north and head towards Cary (200 yards from the hotel). Pass The Ten Ten Rd exit then get off at the next exit to the right, Tryon Rd. You'll be in The Containment Area proper and go to Lucky 32, Southern Kitchen right up on the right. Get the shrimp and grits, jumbalaya, or whatever you want. It's a nice place, but folks show up in every manner of dress. It's a little more upscale, but casual.