Heater blower only on high

Well it could be any one of a few things ! There are two blower motor relays , two fuse's an the blower motor resistor, but wouldn't think that is the problem unless you have high speed blower operation ! But then again don't know if we are even talking about the same fan ? heater fan not cooling fan ?

Sounds like high blower relay is bad

IF THEY CHECKED FOR A DRAW ON THE BATTERY AND THERE WAS NONE DID THEY CHECK ALTERNATOR OUTPUT? IF IT WAS IN SPECS THEN YOU MOST LIKELY HAVE A BAD BATTERY. YOU MAY WANT TO HAVE IT CHECKED AGAIN. THE DEALER IS NOT ABOVE MAKING MISTAKES

I would start with the rear blower motor resistor, the resistor is what controls the speed of the fan, the resistor is located in the rear heater box near the blower, you will need to remove the right rear side panel to access the heater box. start with that

This is your blower motor regulator. It is under the glove box and behind the blower motor itself. They are cheap to purchase but a pain in the rear side to replace. Hope this can help you out sir.

flapper door inside duct work not opening and closing properly. check my tips section for heat diagrams

I would venture to say that if you hit the dash and the motor started running then the problem would be in the heater control assembly mounted in the dash, like bad wire to Heater switch connection. A bad connection can jenerate a lot of heat and start melting things unless you find the problem soon enough. The switch may already be damaged inspect switches and wireing for any vissible damage and replace as required. However if everything looks ok with the heater control assembly, Their are a few more things you can check to be sure of before relpacing the heater blower motor. Turn the heater controles to the high heat and fan on high as well, settings and remove the heater blower motor pig tail and check to see if power is getting to the motor,( use a 12 volt test light at the wire harness side of this connection ) also check for power going to and leaving the blower motor resistor assembly ( mounted to the side or top of the blower motor housing ). If so equipped. If you find no power getting to the motor at this point then you will half to start working your way back through the wireing towords the heater controles until you find the problem. Using your test light. Now if you did have power going to the motor then replace the motor. Good Luck From Ernie @ RedDeer, Alberta, Canada.

This could be a few different things:

1.Check the power and ground connections on your blower motor, corrosion can cause high resistance and a slow blower speed.

2. Check your available voltage at the blower motor on the different settings. it should vary with high supplying 12volts, make sure to do this check with the blower motor still plugged in.

3. Connect the blower to a 12v power source and see if it runs at high speed.

If all else fails i would check your resistor pack, i would not suspect this because when it fails you usually just loose all speeds except for high, but it could happen.

2002 Grand Caravan. The rear heater also has a resistor for the fan speed control.(3 speed) It is located under the rear passenger side plastic cover. You can take off the rear tail gate lower plastic piece and this will let you pull out the side panel. You can reach in and take out the two screws that hold in the resistor and unhook the wiring adapter. You do not have to take off the whole side panel as you can reach in from the back. The resistor unit can be bought at you local dodge dealer.

This is a pretty common complaint and is almost certainly due to a failed heater fan resistor. 
This is a stepped resistor with taps that set the different speeds on your blower and if any part of it goes bad, the whole thing doesn't work any more and the highest speed is the only one that bypasses this part.
These are often mounted on the fire wall but may not be exposed so you really need a service manual (Haynes or Chilton will do) to locate it in your car.