I'm not familiar with Audiovox radios. With many Kenwood and Alpine radios you can press and hold the "Source" button. This causes the radio into deep sleep but still maintains all memory functions and the clock. Long Press on the source is very similar to turning the ignition off on a normal passenger vehicle. It's been awhile since I tested the amperage draw during deep sleep. It might have been about .06amps (+/-?).
Next time you have the radio on. Long press the SRC or Source button to see if this is the case. If so, you could eliminate your switch.