First off, what brand of water heater do you have? I believe the Atwood heaters don't have anode rods and don't need them. Second try flushing the inside of the heater tank, a flushing tool is available from Western Leisure Products and retails for about $12.00(part # TSF 100). By removing the P/T valve and the drain plug you can flush out sediment which as a side effect makes the heater last longer and heat better.
I am neither a plumber or scientist but I have never heard about not drinking water from a water heater tank, in fact, quite the opposite is recommended in emergency situations. We live in earthquake country and it is recommended that folks use the water in the tank if water service is disrupted.
The advice on the thermocouple is very good. I would add that universal replacement kits are manufactured by CAMCO (part #'s vary by length, 12' - 09253) and they run about $9.00 and are easily replaced while cleaning the contact points
Good luck.