To shake hands or to kill is interference and so the two are just the same. Issue then is merely of what kind of time-traveling is being done. If I could meet my grandfather I could meet myself and if I could shake my own hand then I could also kill my own self.
But if I can interact with my own self in the way I can interact with my grandfather then it is clear, two copies are made. How the two copies are made is assumed under the assumption of time-travel.
If two copies are made then my past would be different from the self I meet and so my grandfather would not be my grand father and my self would not be my self. Hence, my interactions with my grand father would not "dissolve" me into thin air because my past is distinct from the one to whom my grand father really relates to or would relate to.
To clarify, my past would include a break - a disjunction. My time line would not be one ray, rather my past would consist of a segment and a ray. The segment would represent my time prior to time-traveling and the ray would would represent my current past, since time-travel.
But if the copies are not made, then time-travel has to be assumed possible only if it leads to some different space. One person in two different spaces at the same time but not on the same time line.