Your question about how capable an AI is for really unsupported multi-day trips is a good one. To me there are a couple of different ways to look at this. Let me first say that I'm no expert on Everglades National Park (ENP), but I have done weekend trips in ENP with both my new AI & in seakayaks. I've finished 3 WaterTribe races, 2 in seakayaks & this years race with my AI, so I do have a feel for what works for me to pack for a 4-6 day trip.
Without knowing more about you & your plans, I would first try to keep in mind that we are talking about doing a weeklong trip with a 16'SOT, would I think of doing this type of trip in say a Tarpon160? I would, but it would require alot of displine with packing & route planning. IMO, a 16' kayak is the shortest length I would go with for this type of trip (5-7 days TOTALLY self supported). You are right with the fact that although the WaterTribe races cover 300-320 miles in a week, most of the trip I was never more than 1-2 hours away from a fresh water source (might be somebody's boat dock) or a 1/2 day from food. When I used a 16.5' seakayak I was load to about 80%-85% capacity, with a little more experience, with a 18' seakayak I was maybe 70% loaded for a 6 day trip. My 18' seakayak was a real gear hauler & I felt I could have done a 14 day trip. With my AI I was loaded to about 80%-85% knowing that water & food was not really a big concern, but I did leave my small camp chair behind, which goes below deck for my weekend trips.
My feelings are that one can pack alot more in an AI than you might think. Assuming that you would be sailing as much as possible, you can pack alot on the rear tankwell. more than you would do with a seakayak. For my longer multiday trips I rig up tramps on both sides. It would not be a problem to think of carrying 2-3 gallons lashed down on the tramps, in fact could be used for a little bit of ballast on the windward side. I would carry smaller water bottles, but would try to carry most of the water in soft sided "water bags" that would lay flat against the bottom of the hull.
IMO, it would just be prudent kayaking to use drybags/dry boxes for as much as you can. Even food lables can be hard to read after being wet for 3-4 days. So yes, just expect your stuff to get wet even inside the hull.
My other thought is that multi-day trips in ENP are not to be taken lighty. A 3-4 day trip even with good weather & fair seas can wear me down while in ENP. Always in the back of my mind I'm trying to always think about what Plan B will be if ...... a raccoon gets my water, my GPS (and backup GPS) stop working, I step on an oyster shell, get a catfish barb in the hand, I lose my hat/sunglasses, get lost, get stuck on the wrong tide, have to portage thru 100yds of knee deep mud, and the worst ..... running out of medicinal rum ... I really hate when that happens.
Another thought to consider is heading North instead of South from Everglades City. The 10,000 Islands around Goodlands/Marco Island are much different that the area of Shark River Etc. Hard bottom is easier to find the chance to beach camp is easier and supplies are closer without giving up the sense of being on your own. Cape Ramono is one of my favorities.
One last thing I've found out. With a little bit of work (and an anchor) it is possible to sleep overnight on an AI, I've had to do that twice in ENP because of low tides did not make it possible (or at least a good option) to reach a campsite because of knee deep mud. As I said, just try to always think of Plan B & don't run out of rum. Good luck.
Johnny Molloy's Paddler's guide to ENP is a must read as is http://www.evergladesdiary.com/directory.html