Let me take my idea of what i think are your assumptions as far as I can.
From your question
"The way I understand it, I'll illustrate with a simplified example. Say I am a point particle. At any one moment in time, I can take 2 states. Then time progression would be a binary tree, where any "descendant" of me in the branches would be different from the root of the tree (original me).
And as the comments suggested, this is actually the same question as "can I keep flipping heads forever"."
You want to achieve immortality by using the splitting process of the MWI to preserve your body. Each copy must remain "you". So each time there is a split in the conditions of the particles that make your body , it must preserve the conditions exactly before the split (so the copy is still "you".) This is my (maybe wrong) extension of the idea of getting "heads" in each toss. A continual set of heads means, in effect, no change and no change to me, means no time passes.
Now if I'm wrong here in my understanding of your explanation, I apologise and what follows is suspect.
What i will be trying to say is that the appearance of a series of heads appearing in our world is equivalent to a series of frozen time universes withinin the multiverse.
If no time passes, (my extension to the multiverse of your unending series heads, implying no change) how will you develop in the first place?. To get from cot to zimmerframe, your body needs to change, but to preserve the conditions immediately before the split to keep the copy the same "you", (you want to keep getting heads) means no change, so no human development even starts. There is to me, a complete paradox here already.
Now say you start splitting at age 25. (and you (or your copy, sorry) still want to live forever. Again, this paradox arises. If all splits preserve the exact conditions before they happened , then a series of universes frozen in time will be produced.
this is probably covered a million times before on this forum, i havn't looked (and I am still not sure that i have understood your assumptions correctly).
I don't think you could achieve immortality by using your assumption But I do believe its a very good question for following the logic/ paradoxes inherent in the MWI idea.
Anyway even if you started splitting at age 25, a set of frozen in time universes ain't much fun to live in.