Short story: fixed it with about $20 worth of parts and an hour or two of work.
Long story: it took many hours/days of research, hypothesizing, testing, going to various auto supply stores, waiting for parts, etc.
After researching various websites and familiarizing myself with the various parts/connectors in the van (removed doghouse; dash where glove box would be; popped hood; got under van; etc.) I bought a hand vacuum pump/gauge (~$40) at Pep Boys, and started disconnecting the vacuum hose from the intake manifold/throttle body and testing vacuum. Well, actually, I couldn't disconnect it at the intake manifold - it was just too difficult to pull it out of it's rubber boot and I didn't want to break anything, so I disconnected it at the rubber boot connector just behind the oil filler cap (note: the black rubber boot and yellow plastic connector is the original vacuum hose that I disconnected; also note: all the pictures are rotated ccw 90 degrees - not sure why):
I attached the vacuum pump and checked if the climate control system (i.e. check valve, vacuum canister, vacuum tubing, climate controls) could hold a vacuum - it couldn't, at least not more than a second or two.
I then capped that line, popped the climate control actuator access panel in the top center of the dash, disconnected the vacuum harness, and tested the main (black) vacuum line looking back into the check valve and vacuum canister. Again, it couldn't hold vacuum for more than a second or two.
This indicated that the check valve and/or vacuum canister were bad. Even though there's vacuum tubing in between that might have been leaking, there were no signs of bad connections, cracks, rodents, so I made the leap of faith that it was the check valve and/or vacuum canister.
I went to the Ford dealer to see what the parts would cost; it would be ~$70 or $80 for the vacuum canister, check valve, and additional vacuum tubing. I also got the impression from researching, although no really good explanation, that the check valve and vacuum canister replacement was labor intensive. I asked Ford what the cost would be; the service mgr talked to the tech and said $933 labor, plus parts, plus likely AC recharge, so I'm looking in the $1100 plus range out the door. Ouch.
Not wanting to pay that, plus really wanting to understand the system, I did some more research and figured out why it is so much labor (there is so much stuff in the way, plus the AC system needs to be dismantled, etc.), and also found some work-arounds. Some people did work-arounds where they rerouted the manifold vacuum hose to after the vacuum canister and check valve, but this doesn't solve the "when climbing a hill" issue. The right way to solve it is by bypassing the E350's pretty much inaccessible vacuum canister and check valve with a new vacuum canister and check valve, but in a much more accessible place. Some people have done this, but they pretty much all did it by drilling a new hole through the passenger side firewall into the engine compartment. I didn't want to do this, especially since there is already a hole where the existing vacuum tube comes through, but I figured that they must know something I don't. In any case, maybe they just didn't try it, but by gently pulling the vacuum hose/grommet out of the hole in the passenger footwell, and then using a coat hanger to fish string into the engine compartment, I was able (eventually, since I was working solo) to following the same path as the existing vacuum tube all the way around and up to the oil filler cap. No need to drill!
I then used the string to route new 3' x 1/8" vacuum tube (~$5) along side the old one, back through into the passenger footwell. I debated a bit then decided to cut out the vacuum hose/grommet and attempt to reuse it on the new vacuum hose. By cutting of the grommet nipple and pulling the old hose out of the grommet, the new hose fit perfectly through the grommet and right back into the firewall.
Now, to replace the vacuum canister and check valve, I found a Dorman 47076 (~$13) which has the check valve built in. I plumbed this in the passenger footwell, between the grommet and the vacuum connector just under the dash in the footwell. I then checked the vacuum from the oil filler cap area again and found that it held a vacuum for at least a couple minutes. Took it out for a ride on a couple of moderate hills and everything worked as it should - vents didn't switch to defrost.
So, took a while to figure out, but the actual work was pretty easy and straightforward, especially since there was no firewall drilling like other solutions required. There's plenty of room to mount the vacuum canister up under the passenger dash but maybe in the future I'll move it to the engine somewhere.
I did use a couple of odds-n-ends pieces and sizes of tubing I had laying about for connecting the vacuum hoses to the vacuum canister, and I also used some various sized plastic vacuum hose connectors.
Hope this helps if someone decides to tackle it.
Most relevant web resources I could find:
Inside vacuum tube connector pictures:http://www.ford-trucks.com/forums/12...-problems.html
Highlights video someone did of the proper fix:
In-depth video of a work-around fix (only a partial fix since he didn't use a vacuum canister or check valve):