At that price, mileage, and age Toyota and Honda are out. However, if you are lucky, you might find a Mazda 6 in your price range and within four years ago. In my area, there is one 2015 for $11,495 with 65,000 miles which is slightly over your range but it is a pretty reliable option if you can find one for about that.
Otherwise you are limiting yourself to an American model at that price point, mileage and age. Best bet? A 2014 Buick Regal. I can find a 2014 Buick Regal Turbo for $11,630 with 67,000 miles on it. Can also find a 2016 Buick Regal Tubo for $11,995 with 33,800 on the odometer. Not loaded but well equipped. Those are fairly reliable and are more likely to find in your price range. They will usually easily last 100,000 miles and more and be fairly reliable (although minor things do come up). it is in my area (CT) but it seems to be a decent car.https://www.cars.com/vehicledetail/detail/751340310/overview/?aff=stemp2&CMP=stemp2
Another car that may work just barely is the Ford Taurus from 2016. it will be near the top of your price range and around 50,000 to 60,000. That is another possiblity that may work for you.
If you liked your current Chevy and it was reliable enough then another Malibu may work for you. Chevy is not quite as reliable as the Buick but it is an okay car. A 2016 at 50,000 miles and under $12,000 should not be too much of a problem.
A website you can use that is the best for searching for used cars is called www.autotempest.com. It searches all the car sites in your area and gives you a consolidated list from all the national car sites. This includes ebay, Cars.com, craigslist, you name it. It is a realatively new site that is excellent for searcing.
Best of luck, let us know what you decide on.