I will apologize in advance, because this is a similar question that has been answered in the past, but I have slightly different circumstances. I replaced the water filter ~3 weeks ago in my GE fridge (3 years old) that has two refrige doors and two freezer drawers at the bottom. This particular fridge only dispenses water at the door, while the ice is available in a tray in the drawer. After installing the filter, the water would stop flowing while using the water filler on the door. This went away after a week or two, but after reinstalling the new filter a couple times, it subsided. Then yesterday, the waterline behind the fridge started shaking violently only when water is being dispensed. Water still flows from the dispenser while the water line shakes behind the fridge. At no other time does the water line shake. I would appreciate any thoughts that may address the issue(s)... Thank you...
It sounds like you may have a portion of one supply line that is frozen, or partially frozen. My GE ended up doing something similar...it ended up being an air diverter valve that the thing forcing colder air where it shouldn't have been under certain conditions.
Important note - I'm not a pro
Retired Defense Industry Engineer; Schluter 2.5-day Workshop Completed 2013