Erratic and observable power-related behaviour is characteristic of backup battery failure in nearly every single iteration of the PowerBooks, though the early ones (1xx series) seem relatively unscathed by it. Duos are certainly affected by it, as are Wallstreets, Pismos and the like. In the 2400c, which is a close cousin of the 3400c, such problems are especially acute. I do not know the 3400c well, having had one only for a few days before passing it on to someone else who needed it, but nevertheless think that verification/ replacement of the backup battery should be item #1 on anyone's agenda if restoration of the thing is what is in mind. In my experience, a functional backup battery tends to have the most dramatic effect on stability and usability, as backup batteries serve a range of functions in a PowerBook that extend beyond as well as including the functions of the pram battery in a desktop.

However, before abandoning hope in the cells you already have, I would try leaving the thing plugged in and charging for a few days, as the backup cells take a long time to replenish.

I do not know what cells are used in the 3400c, but have posted here re. the Varta cells found in the 2400c, which may be the same as those in the 3400c. There's an easy way to find out....