No, no one can get tetanus from washing the dishes.
The tetanus bacteria is found in dirt and spores. The vaccine that prevents it is the Tdap, which includes a larger dose of tetanus, diphtheria and pertussis, which is also known as whooping cough. You need a booster shot every 5 years for tetanus and they always give Tdap, which is painless. The pain in the arm comes from the medication in the muscle. As the medication is absorbed into the body, the pain disappears.