It looks like the problem was actually related to the RAID mode. After tracing a few minidumps and looking at a lot of other blogs, I found something that seemed to work. It may have been trying to boot off the SSD.
1. Set drive mode to AHCI in the BIOS. Boot up the system. (If it doesn't boot, take out SSD, start and shut down, then put it back in.)
2. Shift-reboot, go to "Startup Settings," and reboot, but
3. Stop at the BIOS settings (F2) and set the mode back to RAID.
4. Continue to Safe mode (option F4).
5. Once the system comes up, restart normally.
6. Set up RST again.
Safe mode fixes the drivers it was loading for RAID, even though nothing else is done. After doing this, the system has been running much better. I have the drive reconnected with the connector I thought was flaky and will see if the problem returns or not.