bulletproof knows his stuff.

I'd consider wiring direct from the dimmer to the round-connector terminal that he indicated.

I'd cut the existing wire a few inches back from the round connector and that's where I'd attach a (newly added) lead coming from the dimmer.
That way any modifications to cluster illumination wiring by the prev. owner are eliminated.

While you're in there - make your cluster about 50% brighter this way:

Remove 7 screws holding circuit board to back of cluster, pull off the board.
- [be careful of the hanging trip odometer reset button]
See two reflector chambers for the illum. bulbs.
Glue aluminum foil into there. It will reflect much more efficiently than the dulled yellowed plastic. Keep foil away from the bulb socket area, you don't want it shorting out there. I used contact cement. So far, two New England winters and two hot summers, no problems.

Also while you're in there, do your bit to prevent odometer reset failure:

Carefully pull reset "button" off the little rod it's mounted to. It pulls downwards, not straight out. Look first.
Find flat blade spring that provides resistance when little rod is pressed. Ignore the thin coil spring(s) near rod ends.
Use needle nose pliers to slightly unbend the blade spring so there's less resistance when pressing on the rod.

I've replaced three odo reset buttons.
Relaxed the blade spring every time.
I reset my odo at every fillup, driving 400-800 mi/week.
Never had one go bad again after modifying the blade spring.
Do press gently, once you see how the "button" is built you will wish it was steel instead of plastic. No way can it take significant stress.
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DAMHIK: Don't Ask Me How I Know - - - Sven: '89 245, IPD sways, electric rad. fan conversion, e-codes, 28+ mpg - auto tranny. 500 mi/week commute. '89 245 #2 (wifemobile). '90 244 (spare, runs).