You cannot establish priority without circulating your paper widely. Or, more precisely, you can prove that you had the idea first, but that won't help you much. Academic credit is awarded for contributing to the research community. If you have an idea and don't publicize it (so few people find out or learn from it), and someone else rediscovers the idea and tells everyone, then they will get most or all of the credit, because the community will have learned far more from them than from you. Even if you can prove you made the discovery first, you may end up as a footnote or side comment in the background sections of future papers.
This is not always fair, but it's a pretty good system. In particular, it avoids the nightmare scenario of researchers trying to establish priority while deliberately telling as few people as possible (to preserve their head start for follow-up work).
The fact that the arXiv has time stamps and keeps old versions available is potentially useful for resolving disputes, but the important aspect is that papers on the arXiv get noticed. That's why the arXiv is a good way to establish priority: the community doesn't consider "Oh, I had no idea that paper was on the arXiv" to be a good excuse. By contrast, if you post the paper somewhere much more obscure, then not noticing it would not be surprising or frowned upon.
How to handle this depends on the threat you are worried about. If you are concerned that someone you ask to look over the paper might steal your ideas, then you could cc a trusted mentor who could vouch for you if there were a dispute. On the other hand, I wouldn't worry about this too much: outright theft of ideas is rare, and why would you even want advice from someone you think might plausibly be a thief? If you are concerned about staking out credit in case someone else is independently working on similar ideas, then there's nothing you can do except get your paper in good shape and then distribute it.
If my paper is wrong and I put it on arXiv, I'll look like a crackpot. I want something a lot more obscure than arXiv while my paper is in its formative stages.
Of course it's hard to say without more information, but these comments suggest to me that worrying about credit may be premature. I'd recommend revising/polishing the paper until it is no longer potentially embarrassing or in a state of flux, and then posting it to the arXiv.