So I checked into it. It looks like that the Buick Century had a problem with the current limiting resistors for the lights inside the odometer & shift indicator. If you can solder, it will save you a ton of money. Used instrument panels can run $50 to over $100 on ebay.

Before you go all out and start this repair, double check the fuse in the panel. There was one video which the fuse was corroded and causing the issue.

Resistors don't normally go bad, but the engineers that designed that panel may have made a mistake on the type of resistor to use. It looks like these resistors are getting too hot which is an indication that these resistors are not the right size to handle the current. I wouldn't change the resistance, but a higher wattage rating should be used to replace.

I've repaired many circuit boards in my time, and this would be one of the easiest repairs to do. If you think you can solder and have the time, then I recommend to try it. If you replace the surface mount resistors with axial resistors (ones with leads), I recommend to get some silicone sealant to "stake" the resistors. You don't need to encapsulate the resistors, but a tiny dab between the resistor and the board will help prevent vibrations from breaking them loose. Keep in mind that the resistors may run a little warm so don't completely cover them in the sealant. Normally, axial components have leads that are placed through a hole and then soldered. This gives the parts sufficient mechanical strength. Since these circuit boards don't have a through hole design, the vibration and leverage on the solder pad can cause the pad to delaminate from the board. If the resistors have been overheated, then the solder pads can be compromised and susceptible to delamination. Use plenty of solder flux to solder the parts which will help the solder flow and wet. It is very easy to clean any flux mess with rubbing alcohol.

Before you put any silicone on the parts, you will want to clean any residue. Use isopropyl alcohol and a small brush. Rubbing alcohol will work, it just has much higher water content and will take longer to dry.

Good luck and I hope it saves you some cash to spend at the pub :-)