OK first that diagram is wrongly labled. The YELLOW wire does not go to the start relay
It is correct that with key on (and you should not leave the key on) the ballast will get hot
Check if you have power to the coil+ when cranking on the starter. If not you have disconnected the brown ballast bypass circuit. This comes from the key off IGN2, usually brown, goes into the engine bay, and connects to the coil + side of the ballast. It should have been present with your original wiring, you likely cut it
UNLESS...............I don't know when / if Dodge went to the newer style "Jeep" starter relay. Look at your starter relay........how many terminals does it have? If 4, then the above is correct. If 5, then the 5th terminal goes to the coil+ Like below.................has 5..........
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