When I turn on the vehicle the headlights and windshield wipers come on and won't turn off. When I turn the key off there is a delay before it actually turns off and the headlights won't turn off.

Question edited for clarity. Part of duplicate question added.
Duplicate question deleted.
Question moved to model category.

This says Pulse Board or the Switch

https://www.google.com/search?q=2004+Pontiac+Montana+front+wipers+won%27t+turn+off

This says it does have a relay... but

https://www.google.com/search?q=2004+Pontiac+Montana+wiper+relay

Nothing shows where it is located.

https://www.google.com/search?q=2004+Pontiac+Montana+wiper+relay+location

Relay here

https://www.autozone.com/electrical-and-lighting/wiper-motor-relay

It may not be releasing, you can test if you can find it!

https://www.autozone.com/electrical-and-lighting/wiper-motor-relay

..

You didn't take into account the wiper switch is a input to the BCM - body control module ,or do you know that ? The BCM controls wiper one an wiper two relay's to turn on the wipers .
Wondering if it is the PCB or perhaps a physical problem? Guessing never finds the problem . Reading how they work in factory service repair info should be the first step for someone that doesn't know .
Intermittent Operation
Windshield wiper DELAY operation is a low speed wiper motor function with a variable delay interval between the wiper motor cycles. The DELAY interval is determined by a series of 5 resistors within the wiper/washer switch. During DELAY wiper operation, the signal circuit voltage is further reduced through the DELAY resistors. The body control module (BCM) monitors the voltage on the signal circuit to determine the delay interval between the low speed wiper motor cycles.
If the wiper switch is moved from a longer delay interval to a shorter delay interval, the BCM will command an immediate wipe cycle and reset the delay timer to the shorter delay interval.
Additionally, intermittent wiper operation is vehicle speed sensitive. As vehicle speed is increased or decreased, the wiper interval will also increase or decrease.

Low Speed Operation
When the wiper switch is in the Low position, voltage is supplied through an internal resistor to the signal circuit. The body control module (BCM) monitors the windshield wiper switch signal 1 circuit and energizes the wiper 1 relay which will operate the windshield wiper motor. The wiper motor will operate at low speed, since the wiper 2 relay is de-energized. Low speed can also be commanded with a scan tool by commanding Wiper Relay Cmd. ON and OFF.
High Speed Operation
In the High position, the windshield wiper/washer switch supplies voltage to both the wiper 1 and wiper 2 relay coils. This energizes both relays and allows battery voltage from the wiper 1 relay switched output to the switched input of the wiper 2 relay through its contacts and out the switched output of the wiper 2 relay to the high speed circuit of the windshield wiper motor. The windshield wiper motor will operate at high speed. During HIGH speed wiper motor operation, the wiper switch signal circuit is at the LOW speed signal voltage, and the body control module (BCM) supplies redundant voltage to the wiper relay via the wiper diode.

Wiper/Washer System Components
The Wiper/Washer System consists of the following components:
• Windshield wiper/washer switch
• Body control module (BCM)
• Wiper 1 relay
• Wiper 2 relay
• Wiper diode
• Windshield wiper motor
• Windshield washer pump
• Windshield washer nozzles
• Windshield washer solvent container
• Windshield wiper arm and blade assemblies
• Windshield wiper transmission
• WIPER/SW 10A fuse
• WIPER 25A fuse
Refer to Master Electrical Component List.

DTC B3715 02: Front Wiper Relay Drive Circuit Short to Ground
DTC B3715 05: Front Wiper Relay Drive Circuit Short to Battery or Open

Your best bet , take it to a qualified repair shop.

Sorry , no adjustment .
Wiper/Washer System Components
The Wiper / Washer System consists of the following components:
• Windshield wiper / washer switch
• Windshield wiper system module / motor
• Windshield washer fluid pump
• Windshield washer fluid level switch
• WPR 25A fuse
Wiper / Washer System Operation
The windshield wiper motor module is part of the windshield wiper motor cover and controls the wiper motor operation. The windshield wiper motor is a 2 speed motor. Windshield wiper DELAY operation is a low speed wiper motor function with a variable delay interval between the wipe cycles. The DELAY interval is controlled through a series of resistors within the wiper / washer switch.
Park Position Operation
When the windshield wiper / washer switch is turned to the OFF position while the wiper motor is somewhere in mid-cycle, the wiper motor module will continue to operate the wiper motor until the wipers reach the park position. If the ignition is turned off while the wipers are in mid-cycle, the wipers will stop immediately where they are. The wipers will park the next time the ignition is cycled to the ON position.

Do you know how to do diagnostic's ?

Wipers Blades Do Not Park
Step
Action
Yes
No
Schematic Reference: Wiper/Washer Schematics
Connector End View Reference: Wiper/Washer Connector End Views
DEFINITION: This table diagnoses the wipers operate normally but will not return to the park position when the wiper operation is cancelled.
1
Did you perform the Diagnostic System Check - Vehicle?
YES - Go to Step 2
NO - Go to Diagnostic System Check - Vehicle in Vehicle DTC Information
2
Turn ON the ignition, with the engine OFF.
Move the wiper switch to the LO position.
Move the wiper switch to the OFF position.
Do the front wipers advance to the park position?
YES - Go to Testing for Intermittent Conditions and Poor Connections in Wiring Systems
NO - Go to Step 3
3
Turn the ignition switch to the RUN position.
Turn the wiper switch to OFF.
Disconnect the wiper/washer switch connector.
Do the wipers park?
YES - Go to Step 7
NO - Go to Step 4
4
At the wiper/washer switch connector, measure the voltage of the windshield wiper switch signal 2 circuit.
Does the voltage measure greater than 1.0 volts?
YES - Go to Step 5
NO - Go to Step 8
5
Reconnect the wiper/washer switch connector.
Disconnect the wiper motor connector.
Measure the voltage of the windshield wiper switch signal 2 circuit.
Does the voltage measure greater than 1.0 volts?
YES - Go to Step 6
NO - Go to Step 8
6
Repair a short to battery positive voltage in the windshield wiper switch signal 2 circuit. Refer to Circuit Testing and Wiring Repairs in Wiring Systems.
YES - Did you find and correct the condition?
NO - Go to Step 9
--
7
Replace the wiper/washer switch. Refer to Turn Signal Multifunction Switch Replacement in Steering Wheel and Column - Tilt.
YES - Did you complete the replacement?
NO - Go to Step 9
--
8
Replace the wiper motor cover. Refer to Wiper Motor Cover Replacement .
YES - Did you complete the replacement?
NO - Go to Step 9
--
9
Operate the system in order to verify the repair.
Did you correct the condition?
System OK
Go to Step 2

the dash lights is a lose ground or power connection the wipers could be a couple things there is a wiper relay that may be going bad or the wiper motor is going bad could even be both

Might need to replace the switch

Multi function control switch has to be replaced.

Hi,
The 'stalk' that controls all these functions is faulty. Your truck SHOULD still be under warranty...Take it in.

GL,
Td

replace the COMBINATION SWITCH

2008 should be under warranty, take it back to dealer.

With all of the electronics on vehicles today, seems like these sort of system failures are becoming commonplace.

it is either the multifunction switch (wiper switch) or the wiper motor.