Greetings from another teacher from California - San Luis Coastal.
First, on being in shape: I've started tours in widely varying states of fitness, from being really fit (I thought) to being off the bike for 2 months due to a hernia repair. I've found that it's better to start a tour in shape, but it has never been enough. On every tour the first 2 or 3 days have been tough. Usually the 2nd and 3rd are the worst. I think it's just a matter of acclimating my body and pscyhe to the task. My arms are the worst - especially my forearms.
Knowing this, I now plan very short distances for the first 3 days - 30 miles or less - and consider a rest day after 2 or 3 days. It does wonders to take a day off. Once I get into the swing of touring I average 50 miles a day with occasional 80-milers, and don't suffer too much. I try and take a rest day at least once a week.
So my advice would be to get in as good shape as possible, but don't worry too much if you feel you're not completely fit. And schedule easy days for the first 3, with perhaps a rest day. After that be willing to take a day off if you feel like it.
As to the bike: I have an LHT and really love it for touring. I think the 26" wheels on the smaller frames makes a lot of sense. You can get tires that are very similar to the 700 x 32s I use. I certainly wouldn't consider it a deal breaker. In fact, if you're touring out of the USA, I hear that 26" wheels are much more common and easier to find tires, tubes, etc. for.
You mentioned the Rocky Mountain tourer. I've heard nothing but good things about that bike.
About Spandex: Suit yourself. You sound exactly like my wife. She also refuses to wear Spandex. She has some mountain bike shorts with padding that she loves. I got a gift certificate to a bike shop as an end-of-the-year teacher present, so I bought some baggy mountain bike shorts similar to hers. The padding is nice. They have pockets (?) and a belt, but I still wouldn't wear them on tour. The baggy pantlegs rub on my legs with each pedal stroke. I think after a few 80-mile days in a row there might be a chafing problem. That's one reason I like Spandex - no chafing.