I am only a few chapters into the first season, but 12 Monkeys seems to introduce an interesting take on time travel. Fundamentally it is similar to the "instant rewrite" approach, in which the time traveler's actions that alter the past, change the future "on the fly". This happened in Project Almanac when one of the time travelling boys drew a smiley on the neck of his version in the past (just a few hours) and the smiley was being drawn on his own neck at the same time. In 12 Monkeys the main time-traveling character is permanently worried not to alter things that may erase him altogether before he achieves his ultimate goal, which is to eliminate the cause of a plague. He also assumes that once he achieves this goal he should disappear, because in a future without the plague he would not have been rescued from certain death in jail.
Clearly this mode of time travel is one of filmmakers' favorites, as it produces clear and fast effects that can be well narrated cinematographically, but it so happens that I currently think it is the only one that could not really happen as it is logically flawed: as soon as (and I mean "infinitesimally soon") the time traveler has reached the past, the timeline is a completely new one. We can ignore the possibility of him being immediately erased from existence as part of the timeline that no longer exists, as this would just mean time-travel is impossible. So let us assume the traveler remains alive: if he is wearing a watch and finds its version from the past, and scratches its surface, the changes he is making should not affect the watch he is wearing, as it is a different one. In the same way, if his past self is killed - or stopped from being born - he should not immediately disappear, because this past version of himself is another body, another being who may or may not travel to the past, but certainly not the guy who traveled to the past. So the only outcome of this mode of time travel is that someone will become part of a new history in which the effects of his presence make things different from the history as he knew it. That new history is not the old one that is being transformed sort of "live" by the actions of the time traveler, it is a completely new one where events are different and include the time traveler from the outset. In terms of the smiley being drawn on the neck in Project Almanac, the smiley should not appear on the time travelers neck at all - in fact, when the time travelling kids went back to the future they should have met new versions of themselves there - one with the smiley on his neck - and there would be no way to avoid being duplicates of themselves once time travel has happened!
At this point we can decide whether we prefer to think that the new timeline coexists with the old one (as in Continuum), or the old one is just erased and overwritten by the new one. In either case, there is no possibility of seeing the changes happen "on the fly" and for each time travel event there would necessarily be an additional copy of the time travelers: this is clearly the case if timelines coexist, but is also unavoidable if timelines are overwritten (remember that the alternative would be that the travelers are instantly erased together with their original timeline, rendering time travel impossible).
One particular feature of time-travel in 12 Monkeys is that putting objects in physical contact with themselves from the future or the past causes a explosive energy release and quite a bit of destruction.The writers have chosen to introduce it for the sake of the plot, and does not affect the core question of the paradox as discussed above.