I think you’re describing this? https://en.m.wikipedia.org/wiki/Novikov_self-consistency_pri...
Assume that if time travel is possible, then changing the course of events must be also be possible just as directing future events by our decisions in the present is. If one goes back in time and kills their own grandfather, they have altered the future such that they are never born in order to time travel in the first place, a grandfather paradox. However, since time travel must have existed in order for this situation to occur, then undoubtedly someone else at some future date within this timeline will seek to change history as well. To keep this scenario simple, imagine that this person travels back in order to kill you before you are able to kill your own grandfather, thus restoring your original timeline in which you travel back to kill your own grandfather. This is a meta-stable time loop employing a meta-bootstrap paradox. Each timeline exists and is dependant on time travel from the other. Arbitrarily complex structures are possible involving arbitrarily many timelines, which in an infinite meta-time universe suggests the possibility that all time travel must resolve in this way.
I came up with this years ago, but have only ever seen something like it explored once: in the TV show 'Dark'.
Doraemon, a Japanese children's cartoon (1970s to 1990s), created by Fujiko F Fujio, who is actually very well-versed in physics and science, has used something like this as a plot point in several different chapters/episodes and some movies. It doesn't actively call them "meta-stable time loops", since its target audience is children, but the idea and concept is there. Growing up watching and reading this series, I've learned more and more from it as I rewatch/reread old chapters/episodes, every time. I absolutely love it.
Of course, there's no real reason to believe this is a requirement, it's all just an intellectual exercise.
Past you would have no advance memory of meeting future you until future you went back in time and met past you, at which point you would have memories from both points being produced at the same time.
If I remember correctly Dark to some extent relied on time travel to have time travel in the first place, i.e. Claudia Tiedemann needed to go back in time to hand H. G. Tannhaus the blueprints for the time machine that she used to go back in time with. That, in one of the alternate timelines- in the original timeline Tannhaus presumably made a time machine from scratch, or there would be no alternative timelines. In the end Jonas Kahnwald and Martha Nielsen go back in time to alter the course of the original timeline, thus eliminating their own. This is much like you say, except that in the end the original timeline is restored to the point before any time travel could happen, or had a reason to happen. And maybe that's a simpler way to eliminate any paradoxes: no time travel allowed [edit: yeah, sorry, that's Novikov].
Sic mundus creatus est :P
Of course the show ended with eliminating The Knott, leaving the only paradox as being where two people suddenly materialized in the middle of the road from to warn about a bridge being out before just as mysteriously ceasing to exist.