I have a 2002 Saturn Vue v6 3.0 AWD. When I am driving and release the gas petal I get hesitation and a small **** sometimes. Also, it does not want to go into gear normally and will sometimes go up to 3 rpm before shifting to the next gear (mostly when going up hill or having more than 2 people in the car). Is this a transmission problem?

Sounds like you have contaminants in your gas. Buy some STP Gas Treatment in the Red bottle and dump it in your tank. This should fix your problem.

The Saturn Vue came in the FWD and AWD models. As built, for the AWD to work you need the driveshaft. If you remove the rear drive shaft and fix it so there would be no leakage, you could easily drive the Vue as a FWD vehicle.

2002 Saturn ??? What model ? Vue FWD Vue AWD , SL 100 , SL 200 , L100 , L200 , SL , SC1,SC2 , SL1 , SL 2 etc......

2002 3.0 AWD...SES Light and won't accelerate - SaturnFans.com Forums

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Mar 10, 2009 - 4 posts - ‎2 authors SES Light and won't accelerate Vue Tech. ... As in DRIVE it will not over rev when you push the pedal to the floor. ... Does any one have any ideas where I should start? ....2004vue, Vue General, 14, 07-16-2004 12:57 PM ... 99 SL1 Won't accelerate
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Nov 25, 2013
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If You Are Having Hesitation Problems on Acceleration ...
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DONDE SE ENCUENTRAN LAS BUJIAS DE UNA SATURN VUE 2002 AUTOMATICA AWD MOTOR 3.O

Regular gas (87 octane) should be fine for your VUE.

Take care,
Tom

Under the intake manifold,  it will need to be removed.  Not too hard to do for the average diy but recommend buying a repair manual as a guide.  you will need a 1/4" drive T30 torx socket with a long extension one of the bolts is hidden but the others are under the black cover on the top of the intake.  you will also need to remove the coil pack held on with torx bolts.  8 or 10 inverted torx socket needed for that.

the speedo/odometer are computer controlled by the BCM (body control module) and you may need to have The PCM re-flashed !

This is probably a temp thing...night brings colder temps. The wiring may be failing or the steering wheel portion over the horn might harden and cause it to touch the center of the horn to make it go off.
Let me know if this helps.