I plan on doing a bit of grouse hunting next week in the BWCA and to get to where I need to go there will be some water to cross. My question is what kind of gun case would you use to protect your shotgun from the elements and from the bilge water in the canoe bottom. I have a regular cheap case but I'm not sure it would be dry enough to rest on the canoe bottom without soaking up some water.

I would get one of the many floatable cases made for duck hunters. They do not seal up but unless compeletly dunked would be water proof. I would want something that floats for sure. For total protection Pelican makes some great cases. Heavy and costly but real nice. I have one that is 7 or 8 years old that I take when flying to shooting tournaments. Bagged guys have not killed it yet.

I'd go with something like this
gun case
I have one cheaper case like this lined with that egg crate foam and it has worked well for me the last several years. I only use it for hunting while the gun sits in the back of the truck. One thing I learned about these hard cases (from my brother) is to not leave the gun inside the case. About 10 years ago he bought one of the fancy metal sided cases, o-ring sealed and tight latches. He left the gun in the case over the winter and the following summer open the case and his Browning .270 had started developing light areas of surface corrosion. This apparently happened from the case being closed up when there was a lot of moisture in the air. I'm not sure how well he had cleaned and oiled the gun prior to storing it. I think if the gun was well oiled, this wouldn't have happened, however, he has since spoken with other people that have experienced the same issue. So, long story short, don't get one that is completley air tight and leave the gun stored in it. Store it in a safe or a soft sided, breathable case. He has supposedly corrected the problem by drilling several small dia. holes in the top half of the case. He doesn't leave the gun in it long-term either.
Just something to keep in mind. You probably already know this but thought I'd pass it along anyway.
I myself wouldn't be concerned about the elements getting on the gun over a few days in the field. They are made to handle some water. Bring a spray can of oil and a cloth to wipe it down when you are there and give it a thorough cleaning when you get home. If I were more concerned about a soft case soaking up water I would just use some contractor trash bags to put the gun/case in while in transit.

the way to prevent the above is by using a desiccant (those funny bags you see packaged in products warning you not to eat it)

in fact rather than throw them away, save them for uses like this
they are good forever (more or less)

Yep, desiccant will take care of that problem. When I was in the navy we had air dryers that used desiccant. We would get the desiccant in 100 lb drums. Would really suck if we dumped any on the floor as those little balls would go all over the place. I've always wished I would have taken a 5 gallon bucket of that stuff. It would come in handy in my gun safe.