Exciting Toilet question, can you advise? [View all]

For the past 2 weeks or so, the toilet in my upstairs bathroom sounded like it was running all the time. I could see no ripples in the bowl, wiped off the black rubber cap at the base of the refill apparatus. No Luck. I finally gave up and shut off the water.
My son-in-law came over 2 days ago and, after checking things out, replaced the 'Toilet Fill Valve'. This replacement gizmo was made by Fluidmaster. The constant 'fill' sound has stopped.

After the work was done, he told me he adjusted the 'water level adjusting screw' to regulate the amount of water that goes into the refill process. This toilet is a 45 year old American Standard unit that has a very large water capacity in the tank. Saving some of the water is a good thing. It had flushed and filled for what seemed like a long time.

New problem. Maybe this is in relation to the adjustment.
At random times, I can hear what sounds like a 2 or 3 second fill. I can't figure out what is happening. I would like to stop this event. I can't call son-in- law back, he came from a good distance to do the job for me.

Any ideas what is wrong? Any help would be appreciated.