How can we dry out a soaked water heater (and restore a novice plumber's dignity)?
If you have a wet vacuum cleaner with a slotted attachment, use that first to suck up pooling water and very wet insulation. Then pull out as much of the wet insulation around both elements as you can. If it rips, that's OK because you can add it back when it dries out.
If the area around the heating elements and wiring is dry and no water is dripping or running out the bottom, you can reconnect the elements and turn on the breaker and go back to hot showers. Even if the insulation is still damp away from the elements and wiring, you'll be OK. If you're stuffing towels in between the tank and outer casing through the openings where the elements are or doing any type of drying out the inside, turn off the breaker while you're working on it. If you've got a fan you can place in front of the water heater, do it.