Overheating, looks like there is mud in the overflow tank?

I have the car listed above SE model that I bought it a year ago. Within a few weeks it overheated. I noticed it had what looked like mud in the overflow tank. I flushed the whole system and put in a new thermostat. Ran fine until a few weeks ago. Suddenly it overheated again. This time I noticed the coolant looked like brown liquid and there appeared to be exhaust fumes coming from the overflow, a small little amount. I flushed the system again and also added Barrs to it just in case I had a gasket going bad. Ran completely fine while doing the flush and the gasket seal procedure for right at 2 hours. I finally went to drive it and it overheats again. At this point I looked again, the fans appeared to be operating fine. I sent it to the mechanic. They tell me it was nothing more than a bad thermostat. I take it home and the next day I drive it maybe 8 miles and it overheats again. I have it towed to the mechanic. What do I need to tell them to check? Btw I know the heater core has to be blocked, as last year when it got cold, there was hardly any heat inside the car. Thanks