Assuming that souls exist within the realm of physics, how could they or some similar continuation of consciousness continue to exist after death, especially without the afterlife being awful, in a way that doesn't prove that a divine being exists?
Occam's razor is often used to call for the argument "god is not necessary to explain life and universe"
The same Occam's razor you apply above, you can apply to these souls.
Yes, these souls exist. But existing they are just an extension of the universe. if the universe at all doesn't call for the existence of a god, why would these souls make an exception?
So this is a quick summary of Buddhism:
There are no gods.
Buddhists seek to reach a state of enlightment and to to that we follow the teachings of several wise people that have come before us. Life is both endless and subject to impermanence, suffering and uncertainty.
So there you have it, souls, but no gods.
This is more of a philosophy question. As it is, it is one of the largest unanswerable philosophy questions in existence: how do you prove or disprove the existence of a deity or other metaphysical entity. So the answer to your question is simply "you can't." Philosophers have spent thousands of years pondering this question, and the answer is always the same: either "you can't," or that philosopher is selling something.
Ironically, having a soul prove a deity exists is easier than proving one does not exist. While both are still in the "you can't" range, all one has to do is provide one example of a deity existing and it refutes the claim "deities do not exist." To prove that a deity does not exist requires proving a negative. One must prove that there is no possible way for a deity to exist.
To demonstrate the difficulty of this, consider the case of the absent creator, which creates the physical world, then goes on vacation and lets it run until the end of days. This creator then comes back and does something. There is no physical way to prove this is not the case. We may use abduction to come to the conclusion that "there are no deities" is true because it is the "best" hypothesis, but that's a far cry from what you ask.
Indeed, the closest you could get would be if the "soul" provably came from some primordial soul-mass as in Secespitus's answer, and returned to it. Interestingly enough, that is a paraphrasing of the fundamental cosmology of Hinduism, that we are all the Godhead, just divided because the Godhead wanted to be divided, and that one day we will all reunite. It is also at the heart of the greeting, "namaste," which we most often hear from yoga practitioners. The meaning of that is often traced back to the idea of bowing to the "divine spark" in you, and in us all. It is also the structure often found in the druidic religions regarding the divine Earth Mother (e.g Gaia), to which we all return.
So the related question is "how do you define 'divine?'"
There is simply a big pool of soulmass and whenever something is born there is an empty room, which the soulmass tries to fill. It's like a big pool of water and every living thing is like a very, very small reservoir that can be filled with the soul.
This also means that it's not really your soul versus my soul - we are all the same. The same soulmass, just that our bodies are different reservoirs for small amounts of the overall soulmass.
There is no heaven or hell. Just the body being destroyed and the little bit of soulmass going back to the big part. This also means that nobody cares for your soul after death or distributes it to a different body later on. It just becomes one with all the other souls that are currently not distributed.
Does the formation of particles in empty space imply a deity exists? No. Does the Big Bang itself imply a deity exists? No. The only physics that would imply a deity exists is measurable activity requiring a sentient actor to explain it. If it can all be explained as a natural process, whatever you pick as your physics for souls will likewise not imply a deity exists.