My 1997 Buick Century is "overheating?"?

Double-check your actual radiator fluid level and not just the overflow, you can sometimes have coolant in the overflow and still register low in the system. Thermostats do wear out or seals rot, and with a 97 that is possible and not expensive to swap out if that is the case. Next would be a water pump issue, and again the age of the car works against you but isn't a difficult swap and you might be on cycle to swap the accessory belt(s). Some GM products had intake gasket failures, which is a slight possibility but again not a "major" repair. And you could be lucky and just have a low coolant level that caused an air bubble in the system, but doesn't quite sound like it.
Having the system pressure tested isn't a bad idea, that could point out or eliminate a few things, most importantly you want to rule out a failed head gasket on either bank because that can be a major repair or budget breaker.
Only other thing I can think of that isn't a really bad deal is your temp sensor could have gone rogue, but that would also influence the injection system.
So, if you can do the troubleshooting yourself that saves a lot of money, and if you wanted to tinker I'd start with the thermostat swap because its the cheapest route after checking levels and burping the system. Most water pumps will weep some coolant when they start going out or you can research if there are any specific tell-tales for your Buick's pump. IF those don't solve anything you need to do a pressure test.

You have a blown intake manifold gasket.

I bet the radiator fan relay has died. Replace that $15 plug n play part. If the fan comes on when you turn on the AC, that shows the fan works and the main relay isn't working. Also check the radiator fan fuse. Good luck.

There are several possible issues.

One of the most common issues is a stuck thermostat.