The windshield wipers and washer fluid dispenser stay on

Replacing parts just to replace them wont fix a bad wire or part. Better to use a voltmeter and ohn meter to check if your getting power to the wires. You may have a bad connection or bad switch on the column for the pumps. Without checking the circuits it will behard to tell what it is thats wrong. Find a wiring diagram and trace the wires and components.

Check fuses or washer pump. .replace.

Hard to say what's wrong without testing thing's . Is there B+ voltage to the wiper switch ? B + voltage to the wiper motor ? Do you know how to use a voltmeter ? How to test electrical circuits ?
Here is a factory service manual list of componets involved .
Wiper/Washer System Components
The Wiper/Washer System consists of the following electrical components:
• Accessory relay
• Heated washer nozzles w/XA7
• WASH NOZ 10A fuse
• Windshield washer fluid level switch
• Windshield washer pump motor
• Windshield wiper motor
• Windshield wiper motor module
• Windshield wiper/washer switch
• Wiper/washer switch 10A fuse
• Wiper motor 30A fuse

Do you know how the wipers work ?
Windshield Wiper Motor Wiring Tutorial
Wipers Inoperative - All Modes ?????
Wiper Motor Switch Troubleshooting

There is a electric motor that pumps washer fluid . Located in the bottom of the washer fluid tank .
Wash Operation (Front and Rear Wipers)
The front and rear washer systems use the same windshield washer fluid pump. When either the front or rear wiper is in the washer position voltage will be supplied to one side of the windshield washer fluid pump (depending on whether it is front or rear), while the other side of the pump is grounded.
At this point, the windshield washer fluid pump sprays solvent onto the windshield. When the front or rear washer is activated, the BCM receives an input from the switch that is activated. The BCM will control the wipers while in the WASH mode. The BCM will control either the front wiper relay or the rear wiper relay to turn on the wipers. The windshield wipers operate at low speed for 3 swipes and returns to the park position.

When turning on the washers do you hear the motor run ? Have some one try it while you listen at the washer fluid tank ! These pump motors do go bad .
turn OFF the ignition.
Disconnect the harness electrical connector of the windshield washer fluid pump.
Connect a test lamp from the windshield washer pump control circuit to a good ground .
Turn ON the ignition, with the engine OFF.
Press the front washer switch.
Does the test lamp illuminate?
YES or NO !

You have a lvl sensor located in the bottom of your washer bottle, you will either need to repair or change your sensor, or unplug the sensor to keep an open circuit.

It sounds as if the windshield washer pump, located at the base of the windshield washer fluid reservoir, has lost it's seal. When this occurs, washer fluid seeps into the pump and shorts it out. I advise that you remove the pump to confirm this before replacing the fuse. A replacement wind washer pump and fuse can be found at your dealer, local auto parts store or the internet (Rockauto.com has always been a great source for quality parts at a very competitive price and I use them all the time). Hope this helps.

Is your wiper fluid level good?Look for leaks at pump,it's connected to container,also check your hoses leaving pump to windshield,could be loose or binding.

Hood light on ????? hood ajar light ! Could be the switch . But without testing this would be a guess . Do you know how to test automotive electrical circuit's ? Do you know anything about automotive electronic's ? BCM - body control modules ( computer ) The first diagnostic procedure would be to hook up a factory or a professional scanner an check for DTC'S - diagnostic trouble code's in the BCM .
Wiper/Washer System Components
The Wiper/Washer System consists of the following components:
• Windshield wiper/washer switch
• Body control module (BCM)
• Wiper PCB relay
• Wiper high PCB relay
• Windshield wiper motor
• Windshield washer fluid pump
• Windshield washer fluid level switch
• WPR fuse 25 A
• WSW fuse 10 A

Your best bet , take your vehicle to a qualified repair shop. Nothing you can do without knowledge an proper diagnostic equipment .

Rear washer may have it's own reservoir . Can you hear a pump humming when you activate the washer? Look in the under flooor compartment

As it turns out, there is a recall On 2002-2003 Chevy Trail Blazer's
Windshield Wiper/washer
NHTSA No. 04V046000
Recall Date: Feb, 4th, 2004
Call your dealer to see if this is still under the recall to be fixed.
RECALL: 2002 - 2003 Chevrolet Trailblazer NHTSA Campaign Number: 04V046000 www.nhtsa.dot.gov
Manufacturer GENERAL MOTORS CORP. Recall Date: FEB 02, 2004 Potential Number Of Units Affected: 581394 Description VISIBILITY:WINDSHIELD WIPER/WASHER Summary CERTAIN SPORT UTILITY VEHICLES HAVE WINDSHIELD WIPER SYSTEMS THAT WILL NOT TURN ON, CEASE OPERATION WHILE IN USE, FAIL TO TURN OFF/ON BY THEMSELVES (MOISTURE SENSITIVE WIPER SYSTEM), OR HAVE CONTINUOUS WINDSHIELD WASHER PUMP OPERATION BECAUSE THE PRESENCE OF WATER WITHIN THE WIPER MODULES CAN EITHER CAUSE A SHORT CIRCUIT OR CAN LEAD TO THE CORROSION OF COMPONENTS WITHIN THE MODULE. Consequence DEPENDING ON THE OCCURRENCE AND WEATHER SITUATION, DRIVER VISIBILITY COULD BE REDUCED, WHICH COULD RESULT IN A CRASH. Remedy DEALERS WILL COVER THE WINDSHIELD WIPER MODULE VENT HOLE WITH A PATCH. ALSO, THE WIPER MOTOR, CIRCUIT BOARD, AND ELECTRICAL CONNECTOR WILL BE INSPECTED TO SIGNS OF WATER INTRUSION AND/OR CORROSION AND BE REPLACED AS REQUIRED. THE RECALL BEGAN JULY 6, 2004. OWNERS SHOULD CONTACT CHEVROLET AT 1-800-630-2438; OLDSMOBILE AT 1-800-630-6537; OR GMC AT 1-866-996-9463.