From a similar question's answer
If the toilet still drains slowly, then you have a blockage that may
not be solved with a new toilet. It could be an improperly installed
wax seal, perhaps the drain line isn't properly sized, or there's
something that can be snaked out. It's also possible you could have a
venting issue, but I'd expect bubbles, gurgling, or bad smells with
your other plumbing if that were the case.
Water is not magically flowing up-hill from the bowl to the tank...
It is almost certain that the shut-off valve is not... shutting off.
Consider the design. My experience is that the type of valve pictured uses a rubber washer as the sealing surface.
In most applications, a tap (sink, tub, garden hose) carries flowing water rarely. The rubber seal spends most of its life sealing off static water, whose oxygen is quickly depleted.
In a shut-off valve, the reverse is true. Fresh, oxygenated water is constantly flowing over the washer. As the surface of the washer deteriorates, the ability to seal against the seat vanishes.
But the shut-off valve isn't used, so the problem remains invisible.
Until the day it's needed, as in your case.
There's no guarantee that the next shut-off upstream won't fail the same way. I've had to call the City to shut off my water service at the water main, so I could replace the main shut-off, the hot water tank shut-off, and the kitchen hot-water supply shut-off...
And then change the faucet cartridge...
Best Answer
The fill mechanism is worn out and needs to be replaced. If the reservoir fills up to where the water leaks out from the handle then the overflow pipe is too long and needs to be cut down. The newer type fill valves are adjustable for the height of the water in the reservoir. I prefer the fluidmaster but you can choose you own brand.