That much water cannot be produced from the drain cock or the t&p valve without being unmistakably visible. At the top of the unit are the inlet and outlet fittings (hot & cold) which could be leaking, but that volume of water, again, would be already noticed at the top. If the water isn't dripping from one of those pipes and falling into the emergency pan (which, by your description, it isn't), then your unit is shot and has to be replaced.
The warranties on these things is usually six years. It should last at least ten years, so you've had bad luck. Even if you had a ten-year warranty, it'd pro-rated. The labels on the side of the tank would be informative.
With some patience and a strong helper, DIY replacement isn't that complicated.