imo, Spiral Dynamics is the best evidence we have for reincarnation. With time, people become more evolved. And the progression isn't something learned, but something that comes naturally, and is present from birth. People born now are, by default, more advanced than the people being born a thousand years ago.
That said, reincarnation may not be linear. The low hanging fruit the masses like to go for is that we transition from one life to the next in a linear fashion. The problem with this thinking is it stems from the false belief that time exists when, in fact, time is just a product of our "incarnation", and doesn't exist in and of itself.
I think it's likely that souls exist, as there seems to be evolution taking place at a level outside the physical, but that's no reason to assume that we are souls going from life to life. What we are is just awareness, and that's not a soul, nor is it bound to any specific soul. If souls do exist, then we do not own any single soul, and can simply jump between them at will.
So.. do souls exist? yes and no. You are god, you can create a soul and incarnate into a million humans, all with the same soul ..if you wanted. You create the rules, so it's up to you. But, ultimately, no, souls don't have to exist, and so reincarnation doesn't have to exist. The only thing that has to exist is awareness. Everything else is just a product of awareness.