I need to change out a heating element in my water heater. But first I need to drain the tank. My question is: Instead of running a hose from the water heater to drain it, can't I just shut off the main water lead to the tank and run all my sinks and tubs? Would that still drain the tank without running the hose?

No , you must use the hose bib at the bottom of the tank to drain the tank.

Be sure to shut off the power before doing any work on the tank, for two reasons, one for safety working on the wiring, and most important, to prevent the elements from operating not immersed in a full tank of water.Burns out elements every time.

Nope. The tank drain is at the bottom, the inlet and outlets are at the top. Even if you vented the water heater with the pop off valve, the water would only go down to the level of the connection... ie, the top of the heater. Attach a hose and vent the heater. And shut off the power BEFORE you start. And don't turn on the power until the heater is once again full, and water is flowing steadily from the hot water tap(s)

There is a op off valve on the heater which would open in case of a pressure build up. This valve has a small lever which allows you to open it "vent" the heater with this valve.

Missouri hgad the right idea, However you open the T&P Valve,
(see image) to relieve a air lock and not pressure build uo.
As a rule it's that hard working lower element that goes bad. First turn the heater off AT THE BREAKER BOX NOT THE WALL SWITCH. Disconnect one wire from the element and check across the poles with a resistance checker. You should get a reading on the lowest setting, if not replace it. Now as for changing one. With the electricity still off, turn off the cold water supply to the heater and open the nearest hot water faucet faucet to prevent air lock. Now attach a hose to the boiler drain located at the bottom of the tank and drain the tank. If your in a basemrnt drain into a lower floor drain. You already have the inspection p-lates off so after the tank drains we are ready to remove the faulty element. There are two types, flange and screw in. The flange type just take a socket wrench to remove. THe screw type has a large hex nut built in, if you don't have a wrench big enough they sell a element wrench at your local hardware store. Replace with the same type and the same wattage. I use gasket cement when I install the flang type and wrap the threads with teflon tape on the screw in. After you install the element, with the hot water faucet still open turn on the cold water supply. DO NOT TURN ON THE ELECTRICITY YET!! After, AND ONLY AFTER, you grt a full stream out of the hot water faucet can you shut it off and turn on the electricity. Put the inspection plates on, wait a 1/2 hour and jump into a hot shower. Enjoy! TOM