Out of curiosity what was your experience with some of the other rides. I have never been to any Disney before and people online really praise Toy Story and the Tower of Terror. Were those fastpasses all gone very quick?
The Soaring fastpasses vanishing in 15 minutes is a bit of an outlier right now because the ride just opened a few months ago. It's the new hotness.
However, Toy Story fastpasses have notoriously always gone quick. I've been going for a couple years and I would say fastpasses for Toy Story run out within about half an hour of the park opening. Maybe even faster. And that's even with the new Soaring being open. The Toy Story fastpasses still go really really quick.
Tower of Terror you wont have a problem with, and you wont have a problem getting Journey to the Center of the Earth fastpass tickets either. Just get over to those rides sometime in the morning. Preferably Journey to the Center of the Earth first then Tower of Terror two hours later when you can grab another set of fastpasses.
ALL THAT BEING SAID, and I dont know if you currently have plans to visit anytime soon, but assuming you do the best advice i can give is this: don't worry about it. This being your first time to DisneySea I suggest just soaking it in leisurely. It's an absolutely phenomenal park whose best quality is environmental atmosphere. Wander around, enjoy the food. See it both during the day and night. There's genuinely no place like it.
I consider Journey to the Center of the Earth, Tower of Terror, and Sindbad's Storybook Voyage to be the essential must-do rides. Toy Story is fun an all but i find the long lines to be inexplicable. It is sort of an arcade game so I guess that's part of it, but it's not a huge draw for me. I rode it once and never felt the need to ride it again, whereas I have to ride Journey, Tower, and Sindbad every time I visit.
And if you don't happen to have plans on visiting the park soon, you should definitely consider it. :)