The electrolyte doesn't wear out or go bad. It is more than likely that your plates are toast. Replacing the fluid with distilled water would leave you with a pretty useless battery. Without acid, the battery will not function. Topping them off due to losses from evaporation with distilled water works because the acid is still in the battery and you are simply bringing the actual level back up over the plates.
That said...if you want to make the acid safe for transport, you simple need to pour a base into it. The easiest and cheapest way is to just mix it with baking soda. I do not, however, know if it is safe or legal to pour down the drain then. You probably should just contact the local waste managment department.
Just about everyone has an opinion on how to properly test a battery. I work for Batteries Plus so I can give you a quick run down of what we do. A lot of old timers will swear by hydrometers, but the only time we use one is to find a bad cell. We mostly just use it to make sure all the cells are equal. A digital voltmeter that goes out two decimal places is extrememly useful for finding the state of charge of the battery as long as you are sure you are checking it correctly (no draw, no charging, and not a surface charge). The only way to actually test the condition of the battery is with a load tester. A battery showing 12.65 volts might have a 100% state of charge, but but could be bad because it lacks the ability to maintain that charge when a load is on it.
Carbon pile load testers are the best, but are prohibitively expensive for the average person, but certainly are the best. The smaller handheld ones they sell at most parts stores work just fine for most people. If you havea Batteries Plus in your town, you can just bring the batteries there and they will test them for you for free. Just make sure that they are fully charge first because a flat batttery can't be tested.