If I think there is a problem with water leaking from the tank and it's not the flapper, I like to pull the flush valve, sand the inside of the tank, and then reinstall the flush valve.
I have had a fill valve stick before. If you don't have the TOTO fill valve handy, you can use the Korky MaxPerformance on it. The Korky 528T works too.
Normally in this situation, it's water leaking from the bowl that is the problem. I see that the fill valve tube is still on the clip keeping the fill tube from siphoning. That looks good.