Having gone through similar, I’d recommend the severance.
In my experience & what I’ve heard from others’, improvement plans are usually used to justify firing.
In my case, everything changed when I was put on one. My coworkers began to avoid me, the principal (I was a teacher in a crazy wealthy silicon valley city) was always breathing down my neck, & I ended up losing the spark teachers need to be effective.
My case also included disability discrimination which I continue to fight (5 years later) so it might be a bit more complicated. In any event, once you’re identified as a threat to the organization’s stability life there gets real tough real fast. & it doesn’t end.