You do not specifically need an electrical engineering degree to integrate electronics into the aircraft. That's not to say it wouldn't be of use...just not an absolute pre-requisite.
To make the crossover into a sort of systems integration specialist, I would suggest a few alternate steps.
Starting with your structures background means you already have a good portion of the required skills. You likely already know how to generate drawings and substantiate the physical installation of the systems. You also likely took some basic electrical & systems training in your initial degree. For a lot of systems, most of what you will need to do to determine the proper wiring diagrams is to read the installation manual for the equipment you are installing...so you are already most of the way there.
You will need to develop a background in the heuristic aspects of aircraft wiring and aircraft electronic systems. How do you size circuit breakers, bundles? What is appropriate wiring support? What are the proper terminals to use? Grounding? What is TCAS? TAWS? FMS? ARINC 429? DO-160? You will almost certainly learn little to nothing of this in an electrical engineering program. You can learn this in a distance education avionics maintenance program. One suggestion that I know of that is Transport Canada Approved is ICS Canada:
Link
I'm sure there are more specialized or American orientated programs and courses out there...you'll have to look around. Another good source for heuristic knowledge is the standard practices section of the aircraft wiring manual or AC43.13-1B Chapter 11. You will also need to know about EWIS practices. There are a number of FAA ACs related to EWIS and specialty training is out there. In general, the FAA ACs contain lots of useful information related to electrical systems that you will need to review dependent on the systems you are working with.
One key analytical tool you may need to develop is how to perform a system safety analysis. You can take this online through KSU:
Link
Sooner or later you are going to need to increase your electrical chops from your first degree. You will sometimes need to use relays, diodes, etc. to integrate your systems. A good start would be the MITx basic electrical engineering course (it's free):
Link
I would suggest that you continue to look to improve your knowledge overtime both in terms of analytical and heuristic skills.
Finally, the best source for training is going to be your experienced co-workers both in the engineering office and on the shop floor. They aren't going to have the patience to walk you through everything so you have some work to do developing your own chops. But, if you can take on small, simple jobs to start under skilled supervision and work from there, you'll be able to crossover with time and experience.