10 years is easily enough time for a bad dip tube, in fact, 10 years is better than the average for a water heater failure. My guess is a dip tube. But also, has this unit been flushed often or recently, to remove sludge from the bottom? The dip tube is replaceable, and lives under the cold water inlet, available at big box and plumbing stores.. It directs cold water to the bottom of the tank, while hot is drawn off the top. A shower would start drawing from colder water being introduced farther up the tank than normal. But recovery would appear quick.