Sounds like you need that dimmer switch. I had that happen on my 1992 and my sister has had it goe out in her 1991 Park Avenue twice. I did it in my 1992 (about 1/2 hour and $15 for the dimmer switch).
Disconnect battery (at least I did).
Remove lower black panel below steering column.
Remove lower colored portion of dash (1994-newer has a panel to make it easier).
Remove bolts that hold steering column (but DO NOT PULL STEERING COLUMN BACK TOWARDS THE SEAT -use a couple tie wraps to hold the column from dropping too far down as you will be working upside down from underneath the column).
Locate dimmer switch -has the thin shaft that come from the dimmer/signal stalk.
Unbolt that and unplug.
Put new unit in ($15 from Napa or just about any parts store).
Bolt steering column back up.
Put dash panel back on.
Put lower black panel on.
Good luck and let me know if you have any questions -your lucky your car is a Bonneville and not a 1991-newer Park Avenue (1/2 hour job on the Bonneville and 4-5 hour job on the Park Avenue IF you've taken a Park Avenue dash apart before)