Okay. Stop. Let's diagnose.
If the problem is the fill valve (i.e. a runaway fill valve), the water level in the tank will get pretty high and the water will spill into the overflow of the flush valve. When you have the top off the toilet, do you see the fill valve continuing to run and water just flowing into the flush valve and then into the bowl?
You replaced the R528 cap and you still have the problem. So I'm thinking if indeed you have water overflowing into the flush valve, maybe your fill valve is just adjusted too high. You could lower the water level a bit by removing the little locking key at the back of the valve (if it's in there), turn off the water (because the pressure helps keep the fill valve locked in position at a particular height), push lightly down and twist to unlock, then lower the head of the valve, which would make the water level lower. If that's the problem, you're done.
However, I am thinking that perhaps you do have a slow leak past the gasket on the flush valve. Turning off the water may make it stop "running", but a slow leak could still be going into the bowl. The right way to do this is to mark the level that the water is at with a pencil, turn off the water at the wall and leave it overnight. If the water level drops, you have a flush valve leak, and most likely just need to replace the gasket. Let us know if you have the one piece or the two piece Aquia (or tell us the model number), and we can help you with the part number. There are lots of places online to get the gasket for just a few bucks, or it may be available locally.
The 528MP will fit; it's basically the valve you have there. Someone at Korky once told me that the ones they were making for the Aquias have the ability to go a little taller than the 528MP, but it's unclear whether that's necessary or even accurate. I think Terry just uses the 528MP (actually now the 528PRO, which is a version of the same thing) for all the Totos. The TSU99A.X, which is basically a 528MP with a Toto decal on it for most purposes, does fit for sure. Korky also sells the TSU99A.X under its own brand name as the 528GT (NOT the 528T, which is different). Oddly, places online sell the Korky-branded one for like $3 more than they sell the TSU99A.X, so I would go with the former. However, you save about $6 buying the 528MP, which will do everything most people need, so that would be my first choice if you need it.
Candidly, odds are it's a slow leak past the flush valve, because that's what it usually is, but try my recommendations and see.