A bad connector usually causes either smearing to the right of each feature (each character will look smeared) or quick flashes on the screen, usually affecting one whole color (picture becomes tinted). What you describe seems like either the degaussing is stuck on (but you should see rainbows, not just black or grey), something wrong with the picture tube (shadow mask - degaussing may help, but it may have shifted), the internal power supply has done bad, or there's something near the monitor causing interference (speaker).
You can check for the degaussing being stuck on by temporarily disable the degaussing coil, but don't do this unless you know how to avoid electrocution. Turn off and unplug the monitor from everything. Inside, around the front of the picture tube, will be the degaussing coil. It may be wrapped in electrical tape or a vinyl covering. Two wires lead from it to a circuit board, and unplug it from the board. Reinstall the back of the monitor, and if the rainbows are now gone, you know that the degaussing control circuit has something wrong with it, probably a bad relay or transistor driving the relay.