Plumber told me it's unsafe to drain my water heater -- what should I do?
Ranch experience says, shut off the power and the cold water inlet valve, drain the thing and leave it sit empty.

Get out your biggest Dutch Oven, fill it with water, heat it and take sponge baths. If you're out of clothing, go to the nearest laundromat and process your smalls and whatever else you're running out of.

Whatever this "Home Warranty" thing is, as "The Evil Greebo" points out, you probably have other coverage that will be more useful in a timely manner. 

And find a plumber that isn't going to hand truck a full tank of water down stairs when you could run a hosepipe out the nearest window and water the lawn.

Make sure he's licensed, bonded and insured in case you decide that "50 Gallon Air Drop" is a viable way of removing this water heater.