The running lights are a seperate filliment/circuit in the tail lights - the brake and turn signals are a different circuit. My first try would be to change the problem bulb which has two lights in it (one for brake one for running). If that didn't fix it I'd look for a connector problem with the wire that goes to that bulb or the bulb socket. The wire on both sides for the running lights will be the same color, the brake/turn signals will have different colors in left and right side. Idealy you should put a volt meter on the bulb socket and check for 12V. Since the other side running lights work there is not a fuse problem (probably).
I had an Olds back in the 70's that had a fuse blow in the tail light circuit and it caused the interior lights to stay on all the time - had to disconnect the battery until I figured out the cause. Found it by checking ALL the fuses.
06 Commander Limited Hemi - QDII,
Past -89 Comanche, 98 Grand Cherokee (2WD), 2000 Cherokee, 05 Grand Cherokee Limited 4.7
Had to replace the brakes - they opened more gas stations in town
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