If you go with the many worlds theory of time travel, the act of going back in time would result in the time traveller ending up in a branched-off universe in which history plays out differently to the way history turned out in their universe of origin.
This time travel could almost as easily be 'in-person' or 'mental piggybacking', with whatever rules you wanted.
In effect, in the origin timeline, the time-traveller may have ceased to exist as of the moment that they went back in time if they went physically, or the copy of their mind would have ceased to exist if sent back to piggyback on their past self. However, as you are following the post-time-travel branch, what happens in the pre-time-travel branch becomes irrelevant.
Because the post-time-travel universe becomes seperate to the pre-time-travel universe, paradoxes don't happen.
However, don't be too concerned about a situation in a single-world situation. By definition, time-travel to the past may result in looped causality, but you're only going to see and remember a self-consistent end-result.
Let's do a thought experiment:
Imagine that we have a time machine that is a wormhole that can take a small object back in time a few seconds and a few metres distant. The time machine's entry point is set up so that a billiard ball can roll into its mouth.
If the time machine is set up so that a researcher can roll the ball into the machine and see the ball emerge a few seconds before it enters, a ball will only exit the earlier-time mouth of the machine if the earlier ball will roll into the later-time mouth of the machine. Let's say we set up the experiment to catch the ball and roll it toward the later-time mouth repeatedly.
If we let the setup roll the ball into the earlier-time entry mouth of the time machine repeatedly, then try the following tests, we can predict the results:
If we do not disturb or attempt to disturb the ball from going into the entry mouth, the ball emerges from the exit mouth at the earlier time.
If we set up the experiment to try to disturb the ball on its way from the launcher to the entry mouth after a ball has exited the exit mouth, no ball will ever exit the exit mouth unless it fails to prevent the earlier ball from reaching the entrance mouth.
Result 2 happens because the time machine causes looped causality. The ball rolls into the entrance and emerges from the exit, preventing the ball from entering the entrance, preventing the ball from exiting from the exit, which allows the ball to enter the entrance... and so on until something happens to break the cycle. Something will happen to break the cycle, because each cycle will play out in its own independent piece of space-time, with quantum uncertainties ocurring differently each loop.
Depending upon how robust the experimental setup is, the event that causes the experiment to be consistent may vary from trivial to extreme... but from the researcher's point of view, something will happen if they try to cause a paradox.
If it is as simple as the experimenter grabbing the earlier ball once the later ball emerges, the experimenter will fail to prevent the earlier ball from reaching the time machine's entry mouth, or no ball will exit the exit mouth and no ball will subsequently enter the entry mouth. Why? Because of the time loop, the experimenter will only remember the result where the result is consistent.
The more the experimenter 'hardens' the experiment to try to cause a paradox, by building a setup that shouldn't allow for malfunctions, the more loops the paradox will cause until something breaks the loop. The more robust the experiment, the more extreme and unlikely the event which breaks it out of the loop must be. Maybe the ball spontaneously breaks. Maybe a passing car smashes through the wall and breaks the experimental apparatus. Maybe the building spontaneously collapses. Maybe some piece of the experimental apparatus fails... but whatever it is, it will happen, because the experimenter (and the universe) will only 'remember' the last iteration of the loop.
