Are you sure the heads are not blocked. A small amount of dirt in a head can block the water. My lawn had raised over the years with top-dressing, etc., and over winter my grass grows right over the heads so they won't pop up until I cut a hole in the sod for them, but I still get water flow. In your case, it sounds like an internal blockage, as you get flow for a few seconds, probably enough to fill the lines.
I would suggest you remove > Other than trial-and-error digging, are there other tricks or special