Yep, it is more than possible.
(I'm not a pro electrician or plumber, just along-time DIYer.)
You should have a 30 amp double breaker with 10/2-with-ground wiring from the breaker to the water heater (both black and white are HOT...no neutral for 240v).
Turn the water heater breaker OFF and test the wiring contacts on the water heater to make certain that they're OFF.
Replace that corroded element. If that doesn't solve the problem, replace both tstats.
To replace tstats:
1. Memorize or sketch the exact number and layout of screws on the tstats.
2. Get new tstats with the same number and layout (does not have to be same brand).
3. Exchange wires from old to new ONE AT A TIME. Can't go wrong like that.
Should solve your problem.