I will be digging out the schematic for my DX-390 and tinkering with the radio again. This time I will take pictures and share how to modify the backlight to work one of two ways..
1. Replace a component to extend the time from 15 seconds to whatever you like. Such as 15 minutes.. (I currently have this mod in my radio)
2. Add a latch circuit to control the function of the light switch to allow it to be on/off with a press of the button. I am in the process of researching this functionality using a 555 timer or transistors. Please comment if you have any simple circuits requiring only a few components capable of operating on 6 volts.
I may also add information on how to replace the stock, incandescent lamp with a LED while the radio is on my bench. Stay tuned!