I'm assuming you replaced the bulbs with no success. The GM part number (95-96 Riv) for the multi function switch is 26062938. Been discontinued for some time now. Expensive,
http://oemcats.com/oem-parts/26062938.html
If I was you, and I intended to keep the car, I would get the Factory Service Manuals. Working on these cars without one is near impossible. It might not be the multi function switch, the FSM will help you narrow down the cause.
https://www.ebay.com/sch/i.html?_from=R40&_trksid=p2334524.m570.l1313&_nkw=1995+buick+riviera+factory+service+manual&_sacat=0&LH_TitleDesc=0&_odkw=1995+Buick+Riviera+Chassis+Manual&_osacat=0
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98 Riviera SC3800 All stock except gutted air box.
1970 Buick GS455 Stage1, TSP built 470BBB, 602HP/589TQ
Best MPH, 116.06 MPH, Best ET, 11.54
https://www.youtube.com/watch?v=UHCda-t_Jls
https://www.youtube.com/watch?v=sfT2tEO4XcU