it'll be tough scoring a full size sedan for under 10k... unless it's 8+ years old or with 90k+ miles
BMW 3 series 4 door sedans are probably the best used cars. Engines are put together very tight and will last well past 300k miles. They typically come with leather interiors and fully loaded. If you have 10k to spend you can get a really nice one that looks new.
I've had multiple BMWs and I just think their more reliable and les likely to need repair. I really like Acuras as well but you will likely have transmission problems at some point after 100k miles and it's imparitive that you change your timming belt every 100k miles if you want to preserve your engine. I like Lexus as well but I never thought them to be put together as well as BMWs or as reliable.
BMW's more reliable than Lexus? Not sure the numbers back that. I've never had a BMW but my son's Lexus has been in the family from 44k to 149k (10 years) and the only things besides maintenance that has been replaced is a wheel bearing (at around 100k) and an O2 sensor (recently). Though I will add that the heated seats (both and it's not a fuse) stopped working around 100k and we've never fixed them.
I had an e92 3 series and it had nothing but electrical problems. After doing alot of research from owning the car there alot of non engine issues with bmws ingeneral. Great car to drive though.