thejax,basic common sense says that it's ill advised to leave your gun on the beach unsecured for a whole lot of reasons.
Honestly, I think this is one of those situations where legalities aside, it's just not a good idea, regardless of what the law says. Truthfully, if you want a law to point at, it's unlawful to leave a firearm accessible to children. I don't have the time at the moment to dig up the statutes. But, certainly, on a beach, stuffing a gun down into your sneakers and covering it up with your t-shirt to go into the water would certainly violate that law I'm sure.
I see only two choices in this case. If you know you're headed to the beach and are likely to get into the water, either leave the gun at home or someone sits on the beach, in control of the guns, while the other's take to the water.
I suppose you could wear your gun into the water. But, I wouldn't advise THAT either.
Edit: Now that I think about it, if your scooter had some form of a secured, locking compartment, which many do, that'd be OK legally from what I see. You're only risk there is someone stealing your scooter and not only getting your scooter, but, your valuable gun too.
As far as legally transferring to the scooter...just use your t-shirt to conceal what you are doing. I keep a towelin my saddle bag of my motorcycle so I can kind of cover what I'm doing should I need to transfer my gun fromconcealed on my side tothe bag or back again. Takes a little practice and thought, but, not hard to accomplish.