toto drake problem

I've had my drake installed for about a year now and we've just started to have a problem where it will continue to "run" after flushing. Most of the time the water gets up to the line and the valve shuts off, but I can come back later and the water will be an inch below the line and the valve will be gurgling but with no water coming into the tank. To resolve the problem I have to flush it.

What's the deal?

Sounds like it is time to replace the flapper valve. TO verify this, put some food coloring in the tank, then wait to see if it either is gone in the morning (from continued running), or there's some in the bowl. The flappers are available at a plumbing supply and at Lowes.

Another possibility is if the refill hose is inserted into the overflow rather than being attached with the proper clip. This will siphon the tank. The tube MUST be above the overflow and mounted properly.

Thank you, I'll try the food coloring. The fill hose is still installed with the factory clip above the overflow, so I think that part is fine.

So a new flapper did not fix the problem, the tank water is leaking somewhere obviously, just enough to make the valve gurgle without turning on. Where else can it leak? Between the PVC and porcelain under the flapper? There's no water on the floor anywhere.

Yes the gasket under the flush valve or the PVC (flush valve) can have a kink in it allowing water to escape.
You won't see water on the floor as the tank-to-bowl gasket will not let water on the floor but direct it to the bowl.

If you shut the water supply off, does the tank empty? It sounds like it would be, but verify.

If the flapper seat is smooth, no crud on the lip, and it is not being held up slightly because the chain is too tight, or too loose and a link is caught underneath, since you said no water is on the floor, the other place to look requires you to remove the tank. There's a big nut on the bottom of the tank that holds the seat for the flapper valve in place. If that is loose, it could leak around that, and often, it would leak into the bowl, not on the floor.