Somebody made comments and suggestions on my drafts – can I be reasonably accused of plagiarism?
Based on what you said I have the impression that you are the main contributor to this work and he only had some editorial comments. Therefore, you publishing this work, cannot be classified as plagiarism. (If you think his contribution was sufficient for him to expect you to give him some credit (e.g. second author) then you have to be careful because it may cause further complications between you and him. Besides, it is your responsibility to make sure he gets the credit he deserves)

Most of the times people won't actually act against each other in situations like this. I remember a few years ago, I was stuck in a situation where someone decided to publish a team work with his name as the first author, while it was clear to everybody that he was not the major contributor. My adviser sat me down and told me to let it go (and I accepted because he had good reasons). What I'm trying to say is that in cases like this your supervisor would usually mediate and smooths things out before it gets serious.

However, I think the chances of him even trying to accuse you of plagiarism are very slim. But even so, if you publish this work before him, he needs to have enough evidence that he had major contribution to the work. I assume that this is not the case and it is clear from the emails that were sent to you.

I suggest you to simply let your adviser know that you will try to publish this work and don't worry about the PhD student.