Dripping sound in wall after toilet fills

That's a fairly popular fill valve. To open it:

1. Shutoff the water, you don't want a geyser.
2. Lift the float to the top position.
3. While holding up the float, push down on the top cap and twist the cap a quarter turn (I believe counter clockwise).

The cap will then lift off and you can inspect and clean any parts. Before reinstalling the cap, move any small parts to a safe location, get a small cup to place over the top of the valve, and turn on the water supply. This will jet water straight up from the top of the valve and clear out any debris.

Reinstalling the cap is the reverse of the above procedure. Put the parts back, position the cap back on top, lift the float, press down on the cap, and twist.

Note that if cleaning it doesn't work, it's fairly simple to replace this fill valve.

I've run into this same problem several times, and tried the same things you're trying - adjust the ball, bend the arm a bit to put more pressure on the valve, etc.

Ultimately though I've always ended up replacing the fill valve. The pressure from the float ball and arm isn't enough to close the valve completely, a seal is worn out, or something else is similarly out of whack.

ETA: In my experience, the fill valves that don't have a ball but instead have a sort of integrated float seem to be less prone to this problem (like the one pictured below)