Thanks so much for your reply, ksand24. I hadn't thought about the risks of being stuck in the US not being able to travel (or stuck somewhere in between) and the real risk of becoming an overstayer... I don't want to get near those possibilities!
I looked for PSC appointments last night at 00:00, 00:30, and a few times after that when I woke up, but to no avail (no availability at any centre). I'll keep checking, but I'm wondering if they've stopped opening up new appointments with the new system on its way. I really wish there would be more detailed information about what's coming.
I don't know how I missed the fact that traveling while ILR is being processed invalidates the application--I've spent SO much time searching and reading! If I'd known this, I would have booked the PSC appointment when I saw some available earlier. Sigh!
Here's to hoping I'll be able to get an appointment, and if that fails, that the new system will have an expedited option that works for me (and hopefully the system works, period!). If both of those don't pan out, then I guess I'll have to cancel that trip.