Sounds like either your slide got a bit out of round (compressed or dropped perhaps), you haven’t kept it greased, or conversely you kept it well greased and some of that Arizona sand got into the mix. Count yourself lucky, it could have been frozen in a closed position…
Definitely hydrate, especially before trying to put any of the tennon and socket joints together. If the rings have popped off, the wood has almost certainly shrunk. I’d give it at least 24 hrs in a tupperware with a wet sponge, and with the rings tapped back in place, as others have suggested. That may take care of everything.
If it is the metal slide then the real question is why have you been disassembling metal sections at all. At this point I’ve had both antique and modern flutes, all with metal slides. I only take them completely apart at that joint once in blue moon to fuss with the cork if things seem out of wack. And then only indoors in, fairly clean, environs.
I’m sure Terry will have good advice, once he catches back up with us, but I certainly wouldn’t panic. Most likely you will just have to clean and regrease the slide parts, then figure out which axis of the slide cylinder has gone out of round (I’d use a vernier, but a ruler should do). Light pressure with your fingers on that axis, and if the slide edges haven’t been damaged there should be no problem. From what I’ve seen of Terry’s slide designs, they are well engineered, and if you haven’t had an accident, should still fit together. Of course you could always use the heat and ice trick to increase and decrease the diameters of each slide half, but I’d rather know why they didn’t go together smoothly in the first place.
You never said if you got the flute new or used. If used, possibly the first owner might have a suggestion also.
Actually, if it were me, in Arizona, I’d certainly consider switching to a Delrin flute for outdoors playing (or stick with your whistles). Lots of great ones out there, and they certainly are easier to maintain. I’ve got a Seery, with a big voice, that I take when I’m out hiking or camping.