Newsflash . . . Carmax is also a dealer
That said, I’m glad you got $6000 for it 
Buyer beware . . . !
But even if they lose some on this vehicle, it won’t hurt their “pocketbook” much
Sorry you are having this difficulty OP. It lasted ok for 10 years tho, so that’s something at least. You may have just had bad luck, got a bit lemon when you bought it, that’s possible. More likely tho – and this is only a guess — someone working on the car at some point in the past, probably the recent past, made a mistake , didn’t catch it, you drove away thinking everything was fine, and all that caused this to happen.\
Installing another engine or not is up to you On the plus side, . it won’t cost nearly as much as a new car. On the minus side, you’d end up engine of unknown quality in a 10 year old car. The engine mightr be fine, but your transmission might be on its last legs. Or the transmission might be ok for the next 10 years, but the engine and its uncertain engine software compatibility may cause you all sorts of time-consuming problems. I’d probably be willing to take that risk on, but I find those kinds of things interesting, puzzles to solve. You probably just want a reliable car that doesn’t cause you grief. If so, probably best just to buy another car.
The value of your car in its current state? Not much. If you live in Calif or state with a lot of emissions testing standards, you can probably get a check from the State to turn it in to your recycler. They provide incentives like that to get older cars off the road, as older cars pollute more. If you don’t live in such a state, yeah, $750 is about it. No harm done to look up the name of your local auto-recycler in the yellow pages and ask of course. You might could get a little more than that.
Going forward w/your next car, might be a good idea to get some training on how to check-up on the work that’s been done on your cars. Recommendations here for example are for the owner to always double check the oil level on the dipstick after any work involving an oil change, both at the time you are leaving the shop, and when you get home, and again the next morning. For work involving the cooling system, same thing would apply , only checking the coolant level, and carefully monitoring the engine temperature gauge for a few days. It sounds like a big hassle, but actually takes very little effort or time once you are trained how to do it.
It’s over
“We took it to Carmax, they inspected it, found “no mechanical issues” (per their report) and gave us $6000 for it.”
Problem solved
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Pay attention.
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