Most direct way from Savannah to Dallas is I-16 to Exit 27 - GA-358 to GA-96 to US-80 around Columbus to US-280 to I-85 to Montgomery, then I-65 to US-80 West to I-20. Auburn/Opelika would be a logical overnight. Don't stay in Columbus/Phenix City, it's a big Army town and pretty rough around the edges.

Going through Atlanta and Birmingham adds a lot of excess miles and also throws you into Atlanta's nasty traffic.