What you're missing is that it isn't the fill valve. It's the flapper.
I deleted my vitriol at your plumber for the moment. Let's see if we can't get you going again.
The Korky parts are fine. Terry uses them. Some old-timers swear by the Made In Mexico Fluidmaster. Good for them. Others like the Korky, particularly because it's easy for DIYers to install themselves. I have seven toilets in our old house, and every one (from a 1927 model to a recent Toto) now sports the proper Korky 528 fill valve, and they have worked flawlessly for years.
Let's review. The problem is that your toilet is "losing water somehow". Water comes IN the fill valve and goes OUT the flush valve, i.e. past the flapper. The reason the toilet "runs" is that water begins to run past the flapper as the rubber on the flapper ages. It dribbles into the toilet slowly. The excess water in the bowl just runs over the weir of the toilet without "flushing", usually, and you don't really notice it.
Because little bits of water are running out, the tank level slowly falls. When it falls enough, it activates the fill valve for a couple of seconds so it can top the tank back up. That's why it runs. The fill valve is working properly. If it weren't, you would see the water overflowing the top of the flush valve, the part we call the "overflow riser". It's the vertical tube that rises from the flush valve, into which you have connected the refill hose.
It's quite common for people to think that their fill valve is defective when they keep hearing the toilet running. Makes sense: that's the part that is running. But once you understand how a toilet works, you quickly realize that the vast majority of the time, the fill valve is working properly, refilling the toilet because water is slowly running out of it. And the flapper is almost always the culprit. I suspect that tens of thousands of fill valves sold each year in this country are used to replace perfectly-good fill valves.
Like the perfectly-good 528T your so-called plumber charged you to replace.
Let's check and ensure that your problem is likely the flush valve, and most probably the flapper. Go and mark the water level of the tank with a pencil. Now turn the water to the toilet off all the way at the wall valve. (Turn right, clockwise, to shut off.) Leave the thing overnight, or at least a couple of hours. If the water level drops, it's (most likely) the flush valve.
Flappers do wear out. Six years is a good run. But flapper life varies dramatically based on water characteristics, like whether your municipal water is chlorinated. One thing that affects them, too, is whether you use a chlorine bowl cleaner in the tank, and, if so, whether the toilet ever sits for more than 12 hours without flushing (so the chlorine concentration increases as time goes on). Don't know if you use one, but it will shorten your flapper life.
If in fact you have a leak, do this. Go get this one at Home Depot (or many hardware stores or online sources) :
Click this link to find it at HD:
http://www.homedepot.com/p/Korky-3-in-Toilet-Tank-Flapper-for-TOTO-G-Max-2021CM/202666003
Or, get this one at Home Depot or the other sources:
http://www.homedepot.com/p/Korky-3-in-Toilet-Tank-Flapper-for-TOTO-2023BP/205213624
I have the first flapper above in our Drakes, and it works great. However, the second flapper, which is Korky's newest for Toto, is a little stiffer and might last a bit longer in your application. It's worth a try. It says Toto on it, but it, like a lot of Toto's hardware, is made by Korky.
I'm not going to rip on your plumber, but let me politely say that you should use these flappers and ignore the stupid stuff he told you. After all, he charged you for a fill valve
you didn't need, and didn't fix your problem. He may also have put in the wrong fill valve, one that won't properly refill your toilet bowl, but we can deal with that after we fix this.
After you get the new flapper, turn off the water, flush the toilet, and remove the old flapper. Now get yourself a piece of Brillo or similar and scrub the circular area of the flush valve where the flapper met the flush valve. We call that the flush valve "seat". You don't have to scrub it to death, just give it a cleaning. Sometimes, gunk forms on there and lifts the flapper a smidge off the seat, leading to leaks. Now follow Korky's installation instructions. They have excellent videos and stuff on their web site if you need guidance.
Caution: when you set the chain length on the flapper, make sure that it's tight enough to reliably pull the chain into position, but NOT SO TIGHT that sometimes the chain holds the flapper imperceptably off the seat. That happens to people's installations sometimes, and then they wonder why it's leaking. Optimally, you have one link laying flat on the top of the flapper when everything is at rest. If you give it too much chain, it won't pull into flush position reliably, and also you run the risk that some chain can get caught under the lip of the flapper, also holding it off the seat. So one link at rest, two at most.
This should solve your problem. If it doesn't, come back and we'll do some advanced diagnosis. But this should do it. And if it recurs in a couple of months, come on back and we'll do some advanced diagnosis.
A few other thoughts:
If you put the 528T in yourself, you probably properly positioned the refill hose above the lip of the overflow riser, and didn't just shove it down in there. Correct? (Sticking the refill tube down the overflow riser is the other primary way that water can leave the toilet; it gets siphoned out through the fill valve.) If you did it right, then there's no issue. I assume the end of the refill tube is still daylighted above the lip of the overflow riser, and that the tank water is adjusted to shut off 1/4" to 1/2" below the lip of the overflow riser. Correct again?
There is a small chance the that overflow riser is cracked. I have helped diagnose this several times on the forum from looking at pictures, but it's rare. Sometimes, one can find a vertical crack starting at the top of the riser and running down it to some level below the water level. Water seeps through the crack and causes refilling. Usually, it's more dramatic than that, though, so I doubt this is your problem. You can check by pulling the clip for the refill hose off the overflow riser and pinching the riser to ensure that it is solid. This usually happens because the older thicker metal clips can sometimes crack the riser when they are installed or changed. Again, it's rare, but if it happens, it often starts under the clip.
Please update us as to your progress, and we will work with you until the problem is fixed once and for all.
The diagram of the toilet below has a different type of fill valve than yours, the old float-ball ballcock, and it calls the overflow riser the "refill tube". But it's otherwise a good and accurate diagram of what you have. You can see where the "weir" is on the toilet, which is important. You can see how if you add a little bit of water slowly to the bowl, the water level will start to rise and just dribble over the wier. However, when you flush, a lot of water runs into the bowl at once, and a bunch of it goes over the weir at once, filling the trapway and starting a siphon, which sucks the contents of the bowl out, through the trapway, and out the outlet into the drain system. You may be surprised to know that flushes in gravity toilets don't really "push" the stuff out of the toilet; the rush of water starts a siphon which "sucks" the contents of the bowl out of the toilet. That's why trapway design is so important, and Toto's is among the best in the business.
