> But isn't it true to say "that electron state" in my left hand and "that electron state" in my right hand are states of two separate electrons?

They are distinct state, but you can't say that electron 1 is in the left hand and electron 2 is in the right hand. You can say that an electron has filled (or not) the left state and an electron has filled the right state.

> It was my understanding that electrons had a ton of possibilities states/superpositions that they only "chose" one when they became entangled

They stay in superposition until measured, there's no need to bring entanglement into that.

> is it wrong to think of all the possible positions as the "electron" and it's current configuration in my hand as the electron state in this branch of the wave function?

Nope. You can't describe many-particle states by the individual identities of constituent particles as you do in classical physics. When I have three electrons in a bucket, I describe the state of the bucket with electrons in it, i.e., two electrons in the n=1 level and one electron in the n=2 level. But there's no way of knowing which electron is which in those levels.

In the same way, when we're talking about two electrons in your two hands, we describe the state as one electron in the left state and one in the right state, but there's no such thing as left and right electron. This effect is what ultimately leads to things like the Pauli's exclusion principle or the Gibb's paradox, so we know that the electrons fundamentally don't have identities and it's not just a limit of our measurement.