Yup, definitely sounds like the heater valve has a crack in it. If this place can't get parts or cannot do the fix, just tell them to remove the valve and bypass the heater core. This can be done, you just won't have any heat. They can even use the same hoses still in the car. Then refill coolant, check for more leaks, and drive it to somewhere competent. The valve at fcpgroton.com runs for $14, and the two hoses are only $7 each. Add 6 hose clamps, and you should be able to fix this for only around $30 in parts.

Good luck!