eMac Monitor Problem

Alright, I have an eMac G4, I think its 2nd or 3rd gen. Recently it has been having some trouble with its screen.

It boots up fine, but after no more than 20 minutes, the screen goes off. The light doesn't start pulsating and I can't turn it on again without turning off and on again, where I can have another 20 minutes TOPS. This only works for about three to five cycles before it has to sit without power for about a day before I can see anything. Booting in the normal OS (10.4.11) and its resore DVD(10.3.3) bolth have this problem. However, the Hardware Test extended test will find no problem, and the screen will stay on for hours if it is booted from the Hardware Test partition on the restore DVD. It has had this problem for a long time, but repairing permissions on the harddrive and removing/ replacing the backup battery seemed to work temporarily, until now.

When the screen goes off, I can still eject CDs, adjust volume, and it will play the error sound effect if I punch random keys.

I also have another eMac (first gen 700mhz) that works well except for its CD-RW drive. If I can't get the newer eMac working, I would like to salvage as much as possible and upgrade the older one. What all can be transfered? The whole logic board, RAM, HD, Superdrive? Any Combination? I am in need of a little more computing power, and been needing a new CD drive for years...

Thanks!