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Are your keys out of the ignition when the radio 'stays on' ? Are your auto headlights working ?
With the above problems I would search for VSM (vehicle security module) problems which are typical for '02 psd's.
Can you turn off your dome lights with the dash switch? You might have a sticking light switch(es). They are part of the lock assembly inside the door. Try WD40. Also could be the VSM...........
Still sound like the VSM.. There is a current thread on this forum on this/similar issue. One of the guys disconnected the batteries which in turn 'reset' the VSM and everything started to work ok. I never heard about this 'fix' before but you might want to try it. The worst can happen is you might loose your radio stations presets.
In my case I had to replace the VSM ($120) and have it reprogrammed by the dealer ($145 for 10 minutes worth of work!) I installed the unit myself and saved about $200 in labor.
I'd start with the door switches first. mine had similar issues and I pulled the switches out and they were sticky. like previously mentioned, spray w/WD40 work em in a out a little , been working perfect ever since. always do the cheap things first!
Barney