Water Dispenser Starts, then flows very slow.

I have a Whirlpool side-by-side model ED5PBEXTB00, with a through-the-door ice and water dispenser.

When you press the water dispenser lever, the water comes out "normal", but for less than a second, then it drops down to barely a trickle. Initially, the ice dispenser seemed fine, but now it’s not really dispensing ice (little / no water flowing to the ice tray.)

When we press the water dispense lever, or when the ice tray attempts to fill, you can hear a hum. My initial thought was the filter… replaced it, an it was no better.
Searching through the forums, it sounded like it was the water inlet solenoid valves. I disconnected the water connector on the "in" side of valve, and it seemed to have good flow. On the "out" side, I saw the same symptom- a good initial spurt, then a very slow trickle. I’ve now replaced the water inlet valves, but the problem is still there.
What do I check or do next? Or - the refrigerator is 7 years old - is it time for a new one?