Hi Stuart:
I think Nick's advice here is very sound.
There is no requirement that you have to "prove" that the player is cheating. And I think it would end up looking very bad on you if you thought that you caught the person in the act, but the cards that were revealed didn't make the cheat look obvious. If you really are satisfied that the information that you have received is correct, I think I'd rather be inclined to follow Nick's approach and just suspend them - why waste time.
If that is not practical, I also would consider reminding all players at the start of each tournament that even though they may not be professionally-trained shufflers or dealers, there is a minimum standard that they must adhere to, to play. This means gathering cards facing away as Nick mentions, shuffling using a low profile, keeping the deck parallel to the table at all times, cutting the deck, etc. You should also make it clear that not dealing from the top of the deck, or handling the cards in any way that does not ensure a random deal is considered cheating, and is so serious that players even suspected of doing so will be suspended from play at the TDs sole discretion, without proof. My feeling is that the fear of being caught (especially since everybody will be alert to look out for it once you announce it, and peer policing will kick in) should be enough to deter most players, at least in the short-term. The downside of doing this though, is that the overall mood will likely change if people start getting paranoid that everyone else is trying to cheat them! That's why sometimes, it may be best just to suspend the person you suspect and not waste time.
Depending on how bad the players are at shuffling, I would even consider insisting that players (especially those at the tables where you have the "suspected" player) perform a thorough wash before every hand. At least 5-7 seconds, and the dealer must demonstrate to everyone at the table that a thorough, random wash has been performed. This will slow down the game, but you could just chalk it up to certain players not being as proficient at shuffling as they could be, and you simply want to ensure a fair deal.
A difficult situation for sure -- that's why I'm not a big fan of self-dealt games.