The original power supply was the 65w version, for last 14" iBook G4
and also various PowerBook G4s; where the smaller 12" iBook used
a 45w version. So, it is possible with a higher demand of the device
than the power supply can support, there could be contributing issues.
I have several spare power supplies and noticed the differences among
those for iBook and PowerBook G4; the 65w model will work with most.
So, be sure to check the specs on whatever one you end up with against
those of the computer's original supply. I'm always on the lookout for
extra parts that may be handy later, or in the case of a power supply, to
have one with the portable computer and one at the home desk, saves
wear and tear, between both. The replacement available through online
Apple Store is the 65w model. (I have an Apple 65w made in Japan.)
Hopefully the issue is only the AC adapter, and the aging battery.
Good luck & happy computing! 🙂