2012 Mac Pro reaches grey loading screen then shuts down

I have a mid 2012 Mac Pro running High Sierra. This is the second time this has happened - last time I restored from a backup, but that's not working this time, and I'm worried this would still happen again.

The problem

If I turn the computer on, it will make the wakeup noise and present a grey screen. After a few seconds I get a grey apple logo and a loading bar below. The bar moves along for about a minute and makes it 2/3 across. I hear a click from the computer, like a fan or disk turning off, and then a moment later the entire computer shuts down.

What I've tried

Safe mode (holding shift while booting) doesn't work. I can reach the boot selection menu and recovery mode (and EFI boot, in case GRUB can help me). I booted into recovery mode and tried disk utility.

Here's my disks:

• OS X Base System 2 GB disk image - passed a first aid check
• DATA internal disk (this is a HDD I use for time machine, it's in the first HD slot inside the machine) - first aid was greyed out, and it's not my boot disk anyways
• Mac OS HD 959 GB, 20% full external disk (I'm 80% sure this is the internal SSD, but I can't ever find it) - fails first aid check. Here's a few lines I think are relevant:

...

Incorrect block count for file temp716219

It should be 29 instead of 0)

Invalid index key

Invalid sibling link

Rebuilding catalog B-tree.

The volume Mac OS HD could not be repaired.

File system check exit code is 8.

Restoring the original state found as mounted.

Problem -69842 occurred while restoring original mount state.

File system verify or repair failed.

Operation failed...

After this, I tried restoring from backup. Two different disks failed with the same reason:

Validating target...

Validating source...

Could not recognize "/dev/(nul)" as an image file

Could not validate source - Invalid argument

The operation couldn't be completed. (OSStatus error 22.)

Operation failed...

This is where I've stopped.

I have access to another mac (Mojave) and I still have the flashdrive I used to upgrade to High Sierra. I'm not affraid to dive into the terminal or take the machine apart if needs be.

Any direction is appreciated!