Definitely checkout Campisi's Restaurant. Has a bit of history in the area. See Wikipedia and Campisi's website for the backstory. The original location is the one on Mockingbird Ln.

Oddfellows  or anything in Bishop Arts District. If it's nice out, head over to Chicken Scratch - they have awesome southern fried chicken as well as healthier alternatives.. it's all outdoor seating and is connected to The Foundry - a bar with live music outside.  HG Sply Co. is also a great spot, along with many other spots on Lower Greenville. Have fun! 

Adair's Saloon makes a great burger down in Deep Ellum. 

Deep Ellum is the area with "local atmosphere." It is similar to 6th street. The problem with Dallas is we are so spread out so it is difficult to recommend restaurants that would have any kind of atmosphere. Keller's drive in around the lake is cool and somewhat of a throwback in time as well as the Dairy-Ette 50s dinner.

Also for a fun place try The Rustic its a neat grill/bar owned by Texas country singer owned by Pat Green! Think southern food and mason jars! Have fun!

Jonathon's Oak Cliff 

Smoke

Any place in the Bishop Arts District or Trinity Groves

HG Sply Co.

Since you asked for dives with a southern/local flair skip the trendy new expensive joints like HG Sply Co. (Local, yes, old and divey it is not). Check out Babes Sports Bar - as others have mentioned, or Celebration (for dinner), Luckys Café or Barbec's (for brunch), and Peggy Sue BBQ (for lunch). Have drinks at Ships Lounge or the Lakewood Landing.

By all means, you must go to Babe's Chicken Dinner House!!! There are several locations around DFW but a favorite location is north of DFW Airport. Look online for a list of locations. Joe T. Garcia's in Fort Worthis another good bet--a family-owned Mexican restaurant that has been a local favorite for decades!

Nick & Sam's is a great brunch spot! Awesome sangria and bloody Mary bar. Terilli's is also a great local Italian spot for dinner. Very busy at night I would recommend a reservation. If the weather is nice sit on the roof top for a great view!