Lavin's,
Sounds like your fix was the same as mine initially. I too pulled the radio since the thing was driving us nuts.
I really think the radio has a very sensative soft start circuit. As electronics now become more complex, there's no simple power switch on things that completly remove power from them. Being that the radio has a touch switch to turn it on, and also the alarm function that turns it on, there must be power to the radio at all times. The controller within the radio turns it on when needed and unfortunately, it affected by outside interference.
If you're sure it's not caused by the lights, you may want to put a switch on the positive power supply to the radio and manually disconnect power. As you know from pulling the radio, it is connected to the Winnebago wiring through a plug in harness. Looking at the wires, there will be two power feeds. The green/white wire should be left with a constant 12 volt source so you don't lose your clock settings. The orange/white wire however can be cut and spliced to a switch so you can remove the power to the radio when you wish as a last resort to solve the radio coming on by itself.
The wiring diagram is on page 4 of your manual, which if you don't have it, is available at the link listed below:
http://www.asaelectronics.com/techpdf/awm930.pdf
I sort of stumbled on the fluorescent light fix as the turning on was clearly linked to the fixture. When I looked at the lights, I could see that the electrolytic capacitor on the circuit board exhibited signs of failure. Aluminum electolytic capacitors are constructed with two layers of conductive plates separated by an electrolytic. The package is then encased in a can and sealed with two rubber end seals.
Without getting too complicated, there are a couple ways electrolytics capacitors can fail. One is by the electrolytic drying and allowing the internal resistance of the capacitor to decrease thus allowing it to begin to conduct DC. As the capactitor becomes conductive, it shorts out and fails open which is usually indicated by a blown end cap and residue discharged onto the circuit board. In my case, the silver capacitor on the ballast board, clearly had discharged a small amount of electrolytic onto the ballast and the white metal housing of the light fixture.
Just as an FYI, the other ways capacitors can fail is they can shift in capacitance value or increase their equivalent series resistance, which is the internal resistance that appears in series with the wanted capacitance at a given frequency. Neither of those failures is easy to detect and I doubt you'd run into those failures.
If anyone does trace their strangely acting electronics back to their flourescent lights, check the ballasts for signs of capacitor discharge. When I talked to Thinlite about the problem, they asked wheter my refridgerator acted strangely as they had occasionaly had complaints that the failed ballast caused the newer electronic controls on the RV refridgerators to malfunction.
Their fix is to replace the ballast, and the ballast is covered by a Thinlite 2 year warranty. My fix since I was out of warranty is to replace the capacitor. The picture below is the cap I used to replace the stock one. The value was slightly different, but the temperature and voltage ratings of the capacitor was equal to or higher than the original so it shouldn't fail again.