$10K is a relatively fair price. Now if you can find a 2010 for the same money or less by all means buy that. When I bought my 2002 Lincoln Continental with 22K miles in 2009 it cost me $9K. It was by far the cheapest one within 100 miles of me. CarMax wanted $10K-$13K for similar cars with 75K-125K miles!
My car translated today would suggest you can find a 2008 with 22K miles for $9K. One issue I have with my car is the FWD engine orientation. I don't think the Town Car has that issue so you'll be able to service the engine bay a lot more easily. And I don't think they have the problematic air ride suspension either. Wonderful cars. My 1997 lasted to 232K miles. My 2002 Lincoln from 22K to 70K miles needed front brakes and rotors, a left front axle due to a broken tone ring, and a new radiator due to trans cooler leak. In a way, it's almost better to find one at the 50K-70K mile point with brakes, exhaust, tires, radiator, recently replaced and only pay $4K to $6K.
The majority of these cars with under 30K miles are being advertised from $11K to $14K. So finding someone to sell you one for $8K-$9K could take some looking. I'd prefer a 1 owner, non-smokers, not driven in snow type of car. 2006 is already 10 years old. So the age factor can contribute to repairs even if the car has low miles. I'd try not to go back more than 6-8 years unless the price on the 2006 is very good and the condition blows you away. My car has depreciated another 50% from 22K to 72K miles. Worth about $4500 today.
Good luck trying to find a car like you posted for $5K-$6K. The dealers might be able to snag something like that dirt cheap on a trade-in, but the odds that you can are not very good. $5K right now will get you a 2002 with about 50K miles. A 2006 w/50K miles might run you up to $7K.