There are obvious choices looking at used Luxury cars from makes like Mercedes, Lexus LS and Infiniti, but at $22k, you are looking at a high mileage examples that may start needing expensive maintenance.
Thinking outside of the luxury brand box, my mind immediately goes to the Toyota Avalon. They are sort of the best kept secret of the full size car segment. They are basically a Lexus sedan with Toyota badging. Classic old school big sedan feel and good power and luxury features. Toyota reliability. Basically a Japanese Buick.
You might be able to find and Acura RLX in your budget, but you will be looking at 50+k miles on the odometer. Still, these tend to be very reliable and a bit less expensive than a large Lexus while still carrying the prestige of a luxury make.
Something else outside of the box would be a Hyundai Genesis sedan. The 2012 and up models are large luxury sedans designed to compete against Lexus. Recently, Hyundai has since spun them off as their own brand.