Have you gotten a solution yet? I have the same issue and had to jump a wire from my ignition to the running lights just to make them work. I haven't found anything besides replacing the multi switch at the tune of $120 I hate throwing money at an issue without a sure solution.
2006 SE 6 Speed, Solar Yellow. Zone 2" Hydro BB, Sway Links, and Steering Stabilizer. 31x10.5-15 Bridgestone Dueler AT Revo. Alpine CDA9813, Polk MM6, Polk C400.4