x2 on the above. Distributor has nothing to do with the battery.
Battery going dead (not charging) and it not running right could be the alternator. What is the voltage when it is running (preferably using a multimeter at the battery terminals (if not, off the dash)
2003 TJ, 33" BFG AT, 2" BL, 2" BDS spring lift, 55w DDM HIDs