For truly old cars, a Lexus LS is about as reliable as they get. The problem is that you have to get one that's quite old to stay within your limit.
We don't have much information on the reliability of the DTS, and the bit we do have--for the 2006--suggests a repair frequency higher than average, but not awful. 2006 was the first model year.
Others worth a look include the Infiniti M35, Hyundai Genesis, and Chrysler 300. Most M35s in your price range will be the 2006s, a first model year like the DTS, with a reliability stat about the same as a result. If you can find a 2007 at a decent price, it's likely to be more reliable than a 2006. Styling is a bit bland, but the M is roomy and drives well.
You can get a newer car with the other two, the Hyundai Genesis and Chrysler 300. So even though their reliability is probably similar to a DTS of the same age, they'll be younger. With the Genesis, avoid the package that includes the power tilt and telescoping wheel and nav, as many reported repairs involve features in this package.