This is true if and only if we live in a universe with a time continuum that is not branching (i.e. a line). Instead, if we live in a time continuum that can branch (which will result in a directed acyclic graph) the act of traveling faster than light simply creates a new branch in which you go to the past and in that branch your past self never traveled.

This relates to the different interpretation of quantum mechanics. The one that has paradoxes is the Copenhagen interpretation and the one with the branching is the many-worlds interpretation.

There is also the pilot wave interpretation, but I am unsure how to map the example you proposed to that one.