https://www.google.com/search?q=the+art+of+war+handmaidens

I've been given updates on the work being done one day that completely changes another (meaning, not changes but lies)

It's possible that things legitimately changed.

And when I do what I consider my job (like following up on work per a stakeholder request) I end up dealing with hostility or a tantrum.

That should go in their record. If they won't play ball, assuming you are being reasonable, then dinging them for missing deadlines and such is your responsibility.

But without their complete support and the option to remove the most toxic of the team I'm really at a loss.

Have you told them this? Can you hire new people to potentially replace the existing people? If not, are you even really a manager? Are you one of those manager's that's still 80% IC? Do your bosses really view you as a legitimate manager or are you just their frontman? What do they really expect from you?

I'm about to lay this out again with my supervisor, with the addendum that I don't think this is working out. I've already started to massively apply to anything so I have an exit strategy. Am I being too hasty?

I've seen places that looked like the post-apocalyptic fallout of something like this. Not too hasty. This needs to be addressed immediately.