How do I keep a shadowbox card from getting squished in the mail?

Make a box envelope out of cardstock. It's easier than it sounds: cut your cardstock slightly bigger than the closed card, score the edges about three quarters of an inch in on all four sides, then fold and glue those little walls in. That gives you a flat front and back with a gap between them, so nothing gets squished. Trim down a regular envelope's flap or just make the whole thing from scratch with a flap you close with washi tape. Hand cancel it at the post office counter and ask them to stamp it non-machinable if they'll do it, or at least write 'hand cancel' on it. It costs a bit more but it goes in the bin for manual sorting instead of through the rollers that eat anything with depth or lumpy embellishments. Two weeks is plenty of time to make a second one and practice, and honestly the first one always teaches you what to fix.

As for the cover popping open, yeah, that's normal. Accordion springs are doing exactly what they were designed to do, wich is push apart. My trick is a tiny strip of low-tack tape or a magnetic clasp on the front cover, like those thin self-adhesive dots used for closing card boxes. You can also glue a small paper band that slides off like a belly band, which looks intentional and keeps it shut. Don't trim your glue dots or weaken them, because the real issue is the tension in the folds, not the adhesive strength. If the cover still wants to gape, check wheather the accordion strips are mounted too far forward. Moving them a touch closer to the spine gives them less leverage when closed. I've made a few of these and the ones where I got the placement right closed flat-ish on their own, no clasp needed, though I still added a band for mailing just in case.

Seconding the belly band idea, that thing saved my sanity last year. I made a similar pop up card for my mom and the cover would not stay shut no matter what, so I wrapped a strip of vellum around the whole thing and it looked like part of the design. One thing I learned the hard way though: test the band with the card fully closed and let it sit overnight before you commit. My first band was sized to the half open position and it still let the cover gape a little. Also throw a couple of those thin foam pads between the layers inside so the pearls can't imprint on the backing when it's closed.

Weighed mine at the counter last time before sealing anything up, and the pearls plus that little charm pushed it over what a flat stamp covers, so definitely get it weighed with the box envelope on before you commit to postage. I've mailed around a dozen of these book style cards now and the only one that arrived dinged was the one I dropped in one of those self-serve postage machines instead of talking to a human. Also leave the box envelope unglued on one short end until the very last day. Mine always gain a hair of depth once the glue dots fully cure and the accordion relaxes, and you'll thank yourself for having room to slide it in without scraping the crown off.