Maybe it's better if you are a women. Because like so you just always got pregnant by some unknown guy and you can allways pass all you have to your daugther which is you. Like this no one searches for a mother or a father because they all think you do not know the father either.
Also I would suggest to live a really normal live to the outside as it is very strange yet to have no family at all and you want to be as unrecognized as however possible. You should then change your country, but allways to a place where based on your looks one could think you are native. You always have to adapt to the new Ideology and the language of the people which as you see with old people is not a very easy task.
As you have unlimited time to learn new things you could become a surgeon and modify your own face from "Live""life" to "Live""life".
I for me think that to not be found is either close to inpossible if you don't know that your immortal because you will go see a doctor he tells the state and you find yourself side by side with those labor-rats, or not that dificult if you know it because you have plenty of time to plan some sick master plans. And after the twentiest time you do a "Rebirth" it's not such a complicated thing anymore.
Plottwist:
The user who made the question is immortal and tries to find a way to hide his identity ;)