That V6 is one of the better GM engines, the transmission is not bad if you change fluid periodically. The rest of it has the usual GM problems. Here are the most common problems/things to check:

https://repairpal.com/2008-buick-lucerne/problems

1. I worked at the engine plant at Buick that made the 3800 engine. I was an electrician there. They used to bring engines into the plant that they had given the New York taxicab companies. They put them in taxis  for free. they wanted them To keep records of everything on the engines that they did to the engines.  When they changed oil or spark plugs so on.  GM would Have the engines taken out and shipped back to Flint? They had already shipped new engines to the taxi companies so that they could swap engines bring the engines back to Flint and re test them and then they would cut them in segments so that we Could see what the engines looked like and The cam crank and rods and so forth would be laying all over tables so that we could check them out, also the bearings.. one engine had 303000 miles on it. Well? I guess they don't go by miles but it was equivalent to that and you could still see the horning marks in the cylinders. the engines ran great some of the engines hadn't had the oil changed for over 28000 miles (regular oil) And those are miles not hours so it's a great engine. I have a 2003 3800 Buick Lasaber and it runs great 170000 miles.  No problems from the day I bought it in 20 03.
2. I have had 5 buicks 1 with A V8 which was in 1976 and I never had trouble with that and then I had 4 more. Buicks the one before the vehicle The one before the 03 lasaber was a Buick park Avenue it was basically the same engine in it that is in the 2003. I would trust my Lasaber to drive to Florida and back.
3. It uses about a pint of oil between oil changes of about five thousand miles,  synthetic