The iBook G4 may have been (relatively) free of the bad GPU issue, but it still had a fairly common issue: cracked solder joints, typically in the power section. There are some websites out there, readily reached with a good googling.
Anyway, whether simply from heat, stress, or as one of the many consequences of new, lead-free RoHS-compliant solder, sometimes cracks would develop on the solder joints in the power management section. Typically, this resulted in random loss of power or no power, period, but it's possible that it might cause some of your issues as well.
Unless somebody else has a better suggestion (which is always possible), I would advise you (or somebody else) to open that thing up and inspect the solder joints of the components on the board, particularly near the power section. If you find any cracks (you may need a magnifying glass), resolder them and see if that helps.
