Dealer destroys my car

I have quite a situation going on.
I own a Chevy Blazer 1997, which has been fairly free of problems overall, and only has 85,000 (bought it new). However the service engine light came up and wouldn't go away so I decided to get it checked since I was leaving ten days later for a 2500 miles road trip.
I put the car at Teton Motors (Jackson, WY). They found no problem and simply reset the light. They also changed the transmission and transfer case fluids and replaced leaky oil lines. Then gave me thumbs up for my trip although one thing I noticed at once was the car leaking more oil than before, and being low on oil.
I drove the car 2 days, about 20 miles to town and back total and on the 3rd day, after 15 minutes, the temperature gauge suddenly started climbing steadily all the way into the red.
So I put the car back at the dealership the next day, they test drove it and were able to reproduce the problem, figured out that it was a stuck thermostat and replaced it; at that point they realized that the radiator cooling fluid was sludge-like. They flushed the radiator once and gave me AGAIN thumbs up for my trip.
I left the next day, drove 20 miles... and the temp gauge went into the red again. I got towed back to the dealership for $250 (they lent me a replacement car for my 10 day trip), and they wound up changing the radiator. That didn't help as apparently the cooling jacket was clogged up. They've now told me that I needed an engine replacement for $5700, or they would also give me $2500 as a trade-in.
So, 2 weeks ago I had a car with no problems, and now I own a more or less worthless piece of junk.

My questions are:
1. Is there a way to clean up the cooling system without having to replace the engine?
2. Wouldn't a competent mechanic know that if the coolant system was that bad, they should perform more than one flush?
3. Doesn't $5700 seem like a lot? I checked online and found out such engine costs about $1350.
4. Can I trust the dealership to do a good job considering what happened?
5. Do you have any suggestions or advice?

Thanks for posts,
Alan

Sounds to me like they mixed the coolant when they "fixed" your thermostat. Also, $5700 for an engine installed sounds very high. I would try to find another shop or mechanic or dealer or whatever to get another opinion. If the dealer is found to be at fault for mixing coolant, or anything else for that matter, then I would be all over em to get it fixed on their dime. Then again, the dealer you brought it to to start with does not sound very competent, especially since they couldn't read a code. Even if it was not an active code it would have been stored in the memory.

I think I understand what you are telling us in your post, but what I don't see
(or understand) is the reasoning the dealership is giving you for replacing the engine? Can you elaborate?

Cracked block from driving it so hot, perhaps?

The reason the dealer gave me was that after replacing the radiator, the problem remains the same, meaning the engine is clogged up and overheating. He also said that it may be cracked but I don't understand why he would think that as I drove the car about 2 miles with the gauge in the red, before it blew a hole in the radiator. So would 2 miles be enough to crack the engine?