I really doubt you messed anything up. Its hard to, when you're only running the engine for minutes at the time.
- I would recommend doing like a garden hose flush since its still so brown.
- Open up the engine block drain all the way, take out the thermostat, let it flush until the water out of the block is clear.
- Close your engine drain, fill her up with "normal" water. Bring the car up to operating temperature, Turn the heater to hot in the front and back and let it run for a while.
- Drain, refill with distilled, back to operating temp, heaters back on and drain.
Its normal for there to be a lot of bubbling. I squeeze the upper and lower radiator hoses quite a bit to help speed up the burping process. Its really helpful to have one of those burping funnels that fit on the radiator filler neck.
I often park it on an incline so the filler neck is the highest point in the engine bay when its time to burp.
The color of the draining water will tell you where you're at and how long you need to keep doing this.
But think along above lines, the priority is to get clean water from your drain. The water temp. gauge is pretty important, I'd replace that
In the meantime, I'd buy a cheap IR temperature gauge so you know roughly where you're at with temps.
Just keep in mind you have the radiator part of the cooling system, the engine block and the heater cores part to flush. The whole system takes something like 4 Gallons of water/coolant.