You have 2 prolems(unrelated). First step, always check your charging system Prior to Replacing your Battery. Your Alternator should be puttting out about 13.2 to 13.5 volts(some may be as high as 14.2 volts). Check your battery cables. If they have a powder residue on them, clean them with water only.(Not baking soda or pepsi or coke)I've heard it all. Check the plug going to the Alternator and the cables.Make sure they are clean and tight.Dont just look at them, take them off and inspect. Check your Negative cable from the Battery to the Frame or Block and clean it. Have someone with a Volt Meter check battery volts, with meter still attached, start Jeep, volt drop should be no less than 3 volts. With it running, it should show over 13.3volts. Cables and Wiring cause more Battery and Alternator Failures than anything else. Dome Lights are connected to the switch on the side of the Frame so when the door closes, it pushes in the switch and turns off the light(Normally Closed) switch. It could be as simple as a dirty switch. Remove the switch and test. Put your meter on 2k ohms setting. You will see a 1.,which means broken wire. Connect both meter leads together, you will see three zeros, letting you know that you now have a good wire. Now test the switch the same way. If you get different readings, its working, if it stays the same, its bad. Replace with correct part. Good Luck. Help4u
TRY THE #8 FUSE-15AMP BY THE PASSENGER DOOR HINGE SIDE OF DASH. ALSO SEE IF YOU HAVE GOOD CONNECTIONS ON ALL ALTERNATOR AND BATTERY WIRES AND GROUND WIRES TO THE FRAME-ENGINE-TRANS-AND BODY. THIS CIRCUIT IS VERY COMPLEX AND HEAVY DUTY.IF YOU HAD AN AFTERMARKET STEREO,REMOTE START,OR ANYTHING AFTERMARKET INSTALLED RECENTLY THEN START THERE AND DOUBLE CHECK EVERYTHING-SOMETIMES SHADETREES AND SHOP DUDES GET INTO A HURRY OR CUT CORNERS BORROWING 12V+ FROM PLACES THEY SHOULDN'T !!!