Best option from airport to hotel and back. Not really interested in renting a car. Only in town Fri-Sun. Also anywhere good to eat within walking distance of the hotel?

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It's a $40 flat rate for taxis to any location within the Dallas Central Business District
There are lots of dining options within walking distance, or a very short cab drive away. And of course, the Omni has restaurants, too. I haven't seen any reviews, yet, of their restaurants.
I would caution that walking around downtown Dallas is mostly safe, but there are pockets of less traveled routes. Confer with the concierge about what's safe to walk to and what isn't.
The Omni is kind of isolated by the Convention Center. Nearby - a quick cab ride would be the preferred method - Cedars Social and Off The Bone are south on Lamar in the Southside area. Downtown you'll find Chesterfield's, City Tavern, Wild Salsa - all great places to eat.
Oooh, your going to love that hotel!!!