You won't be the only white person, sorry, but this is by far my favorite place to go when I am on that side of town visiting friends and family:
http://www.scatjazzlounge.com/ You will definitely feel like you have stepped back in time, which is why I love it so much, I'm very much into the style of the pre-prohibition era.
*Edit...oops, you said Dallas. My bad! You should still go though
I like Brooklyn Jazz Cafe on Lamar in Dallas:
http://brooklynjazzcafe.com/ Cool place.