Water filling When on cold water, the washer will only fill so far and stop. It won't do anything else. If I turn the knob a little, it will start to drain, but because it's not filled completely, it doesn't wash. It only does this on the cold water. If I was on warm/cold, it works probably until the rinse cycle on cold water. Then it will only fill a little and stop.

SOURCE: Hot water continues to run after rinse, filling tub overflowing

sounds like you are right the water inlet solenoid on the hot side is sticking remove the hoses and clean th escreen out and the valve try agin with a fill on hot to see if it stops on a small load if not replace the water inlet valve

SOURCE: GE washer won't fill with cold water

Cold water pipe had become frozen. FYI this model of washer has a nice feature whereas you can test each component i.e. valves, by setting the timer and reading the LED's on the front panel. Instructions are in the schematic.

SOURCE: GE SpaceMaker washing machine does not stop filling.

try replacing your water selenoid coming in. remember it isnt a mechanical timer it is a computerized timer, so if the selenoid isnt the problem, it might be the computerized timer. computers are very finicky, pain in the a- -!

SOURCE: washer just started on fill cycle to start and continued filling

you have a hose that came loose to the pressure switch or the tub or you have a bad pressure switch
or the fill cup is blocked

SOURCE: won,t drain on rinse cycle

I had this same problem. I replaced the computer and the relay boards without luck. I found it was the FILL VALVES! Apparently when the washer changes cycles it expects water to fill the tub to a certain level. If the valves are not opening and water not filling the tub will cause the computer to just wait and wait and wait and wont advance to the next cycle.