2003 Dodge Neon, when the headlights are turned on the windshield wiper operate all the time. If just the parking lights are on everything is normal, The wipers will not turn off and the delay wiper doesn't work. The wiper will change speeds when the lever is moved. Where to begin look for the issue? Some where the headlights are supplying power to the wipers.
According to the wire schematics, the problem would be related to the wiper/headlight switch (multifunction switch). The lights and the wipers are all controlled by the same components inside switch. There must be a stuck relay, corrosion or shorted wire inside the switch. The multi-function switch cannot be adjusted or repaired. If any function of the switch is faulty, or if the switch is damaged, the entire switch unit must be replaced.
Before replacing the switch inspect all of the wiring to the wipers and headlights. There could be a wire harness that may have rubbed through and is touching the body and grounding or shorting wires. This can also cause this issue. If wiring harnesses check out good then replace the switch.
Changing the switch should resolve the problem. If you need information on the wire schematic or replacing the switch let me know I can help you with the information.