Shutting of running water make toilets run
As a few other posters have alluded but not said explicitly, this is a water hammer problem. In fact several posters here have had water hammer problems. One poster said his problems seemed to get worse when he upgraded to 3/4 inch pipe. That's pretty classic, if you know what you're looking for. Whereas previously friction through the long runs of 1/2" pipe (I presume) limited the flow to sinks and washers just slightly, now with 3/4" trunk pipe the flow is no longer limited by the pipes. More water is driven through the shower and washing machine valves. Although the water velocity through the 3/4" pipe may be lower, ultimately net force equals mass times acceleration. The mass is greater than before, and so too is the deceleration to which the mass is subject now that the valves are fully "in play". Upgrades like this frequently bring inadequate or non-existent water hammer prevention measures to light.
Although the mechanical advantage of the toilet float is great, there is nevertheless a small force exerted against it which is proportional to water pressure and which tends to keep the float more submerged that it otherwise would be without this force. Water hammer hits the float valve and "bounces" the float down for a moment: the valve opens. There is some hysterysis in the system, to keep this from happening due to very small variations in pressure: otherwise all of our toilets would do this all of the time. But this hysteresis also expresses itself as the toilet tank filling for a short time after the major disturbance, which is why the toilet tank does not stop filling instantaneously, despite the fact that the force to which the float valve was subjected is relatively instantaneous. Eventually, a new equalibrium is established: when the tank is so "overfilled" that the float regains the advantage over the pressure pulses, insulating the (toilet) system from further insults of the same magnitude. The annoying behavior stops until the toilet is flushed again. I bet if you sit there at the bathroom sink and bang the cold water faucet open and closed, eventually you will reach a point where the toilet no longer responds.
Unless, of course, the situation is so bad that tank water starts flowing into the overflow tube. If that's the case, the toilet may respond to high pressure insults indefinitely.
In general, cranking the float down won't help unless water is in fact spilling into the overflow tube. Regardless of the float's absolute position at the end of a flush, it is still caught in the balance of the same forces. It may work in some band-edge cases, by "tuning" the system ever so slightly that the symptoms disappear. But generally the toilet is not capable of influencing the forces allied against it.
Suggestions for water hammer arresting abound, including "whole house" arresters and arresters at each fixture. Ultimately you should pursue these types of solutions for the benefit of all of the appliances and plumbing system.
One thing I've never seen mentioned is the idea of an arrester at the toilet. This is because the flow through the float valve is limited in general, and in particular probably slows to a trickle immediately before stopping completely. Thus it is not a common cause of water hammer. However, I would be very curious about the possibility of limiting the insult to the float valve by installing an arrester at the toilet. If the system appears more or less properly "arrested" and the toilet is the last remaining nuisance, you might try an arrester at the toilet. I'd be very interested to know the results of such an experiment.
BTW, the reason I figured this all out some years ago was that the hot water valve in my washing machine eventually gave out and wouldn't pass water. The price of a replacement valve was quite shocking, and I got the impression it was a common problem. So I grabbed a highly regarded industrial solenoid valve from my inventory as a professional mad scientist. The valve was huge in comparison to the original, so I included a ball valve to limit the flow. One day some months later I started hearing water hammer. I figured my home-made arresters had finally filled with water. When I had installed them, I was clever enough to allow them to be easily and completely drained through the washer service valve. I was pretty incredulous when draining the arresters didn't solve the problem. Eventually I figured out that one of my renters wasn't happy with how long it was taking to fill the washer, and had opened my ball valve. I was tipped off by occasional overflows of the soap compartment. Once I cut the flow rate about in half, the water hammer dissapeared. It still seems strange to me, but despite the fact that the arrester was oversized and ideally placed, it was ultimately overwhelmed by the snap action of that electric valve when the water flow and hence velocity exceeded a certain threshold. I had always thought of water hammer as having a broad spectrum: present to some extent in all systems, but not troublesome in a well designed system. But my experience that day suggested this is not necessarily true. It indicated to me water hammer is a NOTICEABLE problem only if a certain threshold is exceeded somewhere in the system. You either have a noticeable problem with water hammer, or you don't. There's no in between.
I'd be interested in hearing thoughts and experiences related to this issue.