On dual flush flapplerless toilets the rods for the button flush mechanism will lengthen over time and keep the flapper mechanism Open slightly. Screw the rod(s) up so that they are not touching the button inside the tank. Once you have the right length and the water flow has stopped apply plumber tape on the threads to hold them in position.
The seal could be bad but when i had the problem i replace the seal two times with no luck. It seemed to work for a while and then the intermittent leaking and refilling started again. After talking with Glacier Bay technical services they helped me get an entire flush mechanism. However something they told me hit home. They said sometimes the dual flush will stick and cause this problem. I then went back in and put a level on the flush and made sure it was completely level. After doing that i have not installed the new flush canister and i have not encountered the problem again.
The flapper valve is bad. Try cleaning it first before you replace it. Good Luck