On one, the screen will only display an image that resembles a lightsaber in the center of the screen
That's definitely a video hardware problem. On the first eMacs. it was the result of a defective IVAD cable. A new one is not expensive (about US$25) but getting to it is best left to the pros, as you're delving into the high-voltyage viper pit.
It can also mean the video section on the logic board has failed. Later USB 2.0 eMacs with 1Gz and 1.25Gz processors had a logic board problem that is not worth repairing; however, with it you usually got no video at all or random crashing.
You may wish to figure out what models you have as there are differences in potential issues and their cures. This Apple article uses the serial number to figure out the model:
If it's ATI graphics, it's not the IVAD cable. Apparently that only affected The first models with nVidea graphics.
This place in Oregon:
has some eMac parts broken down by model variant.