Getting into the aircraft business...
Hi James,
Have you thought about doing some FAA technician courses to convert your broad spectrum electronics engineering knowledge into more tailored avionics knowledge or maybe looking at Embry-Riddle Aeronautical University for some post graduate studies into avionics? As another poster already pointed out, if you were in Europe you'd be a prime candidate for an EASA Part 66 B2 licence.
The OEM's are going through a tough patch right now, but I'm seeing signs of a thaw. Avionics is probably one of the biggest need areas right now as the industry moves towards Integrated Modular Avionics.
If you don't want to work for an aircraft OEM, also consider the avionics OEM's - Rockwell Collins in Cedar Rapids or Honeywell in Phoenix are probably a good place to start looking.
Good Luck with the search
Cheers
FlightTEster