Had a similar problem myself a few years back. Turned out to be a crack in the line. The water was leaking instead of coming out the nozzels as its supposed to be. I just replaced the line. Pulling the lines at the pump to check intput and output as suggested is the place to start. Once you know the pumps okay start on the lines.

I had the same gum-up problem after leaving the windshield washer system in my 79SC untouched for 30 years. Not a drop out of the nozzles. I completely removed the whole system, tank hoses, connectors check valves and nozzles. Cleaned inch individual part and removed a lot of internal jelly-like gunk from every part. Then I re-assembled the whole system on the bench and tested it. It worked as it is supposed to. Re-installed and got the system working again.

Start at the beginning
 

When I replaced my a/c lines I removed the washer tank for access, and discovered why my w/s washers did not work. Years of stagnant water created a scum on the bottom of the tank and blocked the pickup. It wasn't too hard to remove it, dump out the old water and clean it. I tink it was easier than replacing the little check valves.

I had the same problem. Took the tank out and flushed it out. You can't believe the junk in the tank. Also cleaned the pickup in the tank. Works great now!