Mine is an '87, but I believe the wiring for the '89 is same/similar. You will have a purple wire and a black (or black/grey or black/white) wire, both of these run from the trunk area to under the driver side kickpanel. This is your ground (black) and your fuel sending wire (purple). You will find these on the passenger side, behind the back seat (you will have to take the top part out).
I recently had problems with mine, it wasn't working at all, this was due to a bad ground. If you want to see if the ground is the problem, cut the black wire behind the seat and attach it to the frame - there is a ground point there. If the gauge works, then that is your problem.
If not, either the fuel sending unit is having an issue (my aunt's Mazda is doing a similar thing) or the purple wire is having an issue and needs rerun. Since the fuel gauge goes up with more resistance (~90ohm = full), this is unlikely, since it'd mean the gauge loses connection on acceleration. I would think it is the fuel sending unit above all else, though, but the ground is easy enough to check.
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Originally Posted by Gocart879
my low fuel light is stuck on, replaced the bulb a while back b/c it didnt even have one in it. where is the ground for this?
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Mine did the same thing, even with a full tank. From the schematics I have read, I believe this bulb requires another unit (the third pin) to work properly and only turn the bulb on when the tank is low - otherwise it'll always be on. Sorry. :/
Hope this helps you.
Edit: As Dan said below, it could also be fuel sloshing. I also wanted to note that my gauge sometimes also rises a little bit and stays while moving, although not much (if I have half a tank for instance, the needle might go from half to just over half, but isn't based on acceleration, it's based on idle???). I attribute it to Buick-ness or GM-ness. :/
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'93 Dodge Caravan, 3.0L, A670 - It runs.
'85 Chevrolet Camaro Z28, 305 HO, T-5 - Transmission died!

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'87 Buick LeSabre, 3.8L - RIP. Lots of blood, sweat, and tears.