I tried to find out what is in "Clear Water." Here is the site on it:
http://www.junglelabs.com/pages/details.asp?item=NJ021It of course, does not say what is in it! From what they say and a similar product I have at home, I'm guessing it is KMnO4 or potassium permanganate. That IS toxic to snails. The container should say so if that's what it is. See if the paper work that came with it says what the chemical is. If it's KMnO4 the liquid should be pinkish or darker red. What color was it?
Touching the snail gently probably didn't overly harm it. While snails and goldfish can share certain parasites, the chance of something being moved over with the net is very low. Snails are more likely to give something to goldfish as snails seem to be more tolerant of a parasitic load.
Changing water for a snail is just like for fish. Don't change more than 50% at once (except in an emergency), try to add water of similar temperature and water chemistry, leave the snail in the tank if you can during the cleaning, and add dechlorinator if you have city water. Water changes are stressful and may have contributed to the snail's problem. It's rare though that a water change with similar safe water would be deadly.
If the innards of the snail are hanging out and just floating in the current, the snail is dead. On picking it up, if it were still alive, it would hide back into the shell. I'm sorry for your loss.
If the Clear Water killed the snail, the betta should be fine. Bettas sometimes eat snails but I assume you've had them together for a while (and the flesh isn't missing on the snail anyway). What kind of snail was it? I don't think the betta is in danger (but I'm not always right). Good luck!