then that measurement and the effects it has on its local environment and the effects they have are all also dependent on my measurement.
I think that's your trouble there. Those are totally independent of your measurement, the waveform splits at the measurement of the other particle the exact same way, regardless of what happened to you locally.
except I can never be in a universe where it split at my measurement site in one way and it split at the other measurement site the other way. That means that I am entangled with a measurement that happens nonlocally to me and everything that it interacts with.
You're not entangled with the measurement, you're entangled with the entire wave despite the inconceivable number of measurements that happen within it. The same way my friends are still my friends despite us being in different places. I won't know instantly if they die or something, but I react differently when I hear about what happens to them than I do with ordinary people. If one of them gets into a car accident outside my light-cone and I'm more sad when I hear about it than about the car accident of a stranger, that doesn't mean that I have a non-local interaction with the car accident.
I think that's your trouble there. Those are totally independent of your measurement, the waveform splits at the measurement of the other particle the exact same way, regardless of what happened to you locally.