Slow down there, friend.
If it's running, there are two possible reasons. One might be that the fill valve is stuck. This isn't usually the case. The other is that the fill valve is fine but water is draining from the tank and the fill valve is just trying to keep the tank full. That's more likely the case. The way to differentiate is to look in the tank. Look at the overflow riser, which is the pipe that comes up from the flush valve at the bottom of the tank. Is water running over the top of that riser, or is it below there? If below, then it's most likely a flapper issue.
On a flapper issue, it may just be worn out and need to be replaced. Or it may be fine and just not seating properly. Take a look. If everything in the tank looks fine, and it opens and attempts to close cleanly into the hole and just doesn't seal, then you can replace it. Best option: the Korky 3060BP 3" adjustable flapper.
Good luck, and let us know how it goes. (If it does turn out to be a fill valve issue, you can often just replace the cap R528 and fix it for like $3. If you want to replace the whole valve, get the Korky MaxPerformance 528MP (528MPK at Lowe's). The 528T you linked to is good for the original Drake, but will overfill the bowl on the Drake II. You want the 528MP, which is infinitely-adjustable, and cheaper than the 528T, usually.) If it does turn out to be a fill valve replacement issue, come back and I'll point you to a Korky video on how to replace it.