I actually love the Wynfrey myself, but the Sheraton is nice, and if you really don't want to drive, that's the way to go.
It's not HARD to get from the Wynfrey to the BJCC, and of course if you take a cab then it's no worries, but they are doing construction on that section of I-65 and it can be gnarly. (unless you're from someplace like Atlanta or Boston, then you'll think it's just normal traffic.

)
Parking at the BJCC is also, well, lacking. There is a deck, but it can fill up and you're left driving around searching for something on the street, or in one of the surface lots where you have to pay cash in advance. If you're at the Sheraton, you can just walk over. Or the cab would take care of that problem if you stayed further away.
Now, back to restaurants. There is not a ton of choice around the BJCC, but I think Cafe Dupont is not far (8 blocks - walkable for lunch, but I wouldn't do it at night) if you want something higher end.
http://www.cafedupont.net/
If you don't mind driving just a little ways, it's not far, straight up 20th street to 5 Points South, and many excellent restaurants are there, including those of Frank Stitt, a nationally-known chef.
http://www.southernliving.com/travel/south-central/chef-frank-stitt-birmingham-00400000005861/
Highlands is his flagship restaurant, but it's only open for dinner. There is also an Italian restaurant, Bottega, and a French one, Chex Fonfon. All three are EXCELLENT. Check out the menus and see if one is to your liking.
http://www.highlandsbarandgrill.com/
http://www.bottegarestaurant.com/
http://www.fonfonbham.com/
Other higher-end type places nearly are Ocean, and Hot & Hot Fish Club.
http://oceanbirmingham.com/ocean/
http://www.hotandhotfishclub.com/
I haven't personally eaten at these, but they are highly regarded, and someday I'd love to try them both, but somehow we always end up at one of Frank Stitt's restaurants when we have that kind of meal budget - lol.
Birmingham hasn't quite figured out public transportation, but during the day, there is a DART bus (downtown area rapid transit) that could take you from the Sheraton to 5 Points.
http://www.bjcta.org/
Also, if you like art, the Birmingham Museum of Art, is very very close, and it's free. See the stunning "Looking Down Yosemite Valley" by Bierstadt, and there's even a Monet!
http://www.artsbma.org/