iBook G4 shutting itself off randomly

My iBook G4 (1.42 GHz PowerPC G4, OSX 10.4.11) is about 5 years old... about 6 months ago, it started shutting itself off randomly. This happens even when it's plugged in, so not a battery issue. Then when I try to restart, my laptop doesn't respond to the power button until I turn it over, take out the battery & put it back in, and then press the power button again. Usually (but not every time) when I turn it back on, the date & time is reset to pre-2000.
This happened a couple of times, rarely, about 6 months ago, and has since become more frequent - it's happened 3 times this week!
I went to the apple genius bar today and without even looking much at my computer or turning it on, the guy there immediately said it's most likely the logic board, which on a laptop this old isn't worth replacing... I'd very much appreciate a second opinion and/or any advice about things to check out before I give up and buy a new laptop!
Thanks!