Is it safe to show my old advisor my new paper?
If you work in the US or Europe, intend to complete the paper and submit it for review somewhere soon, and have records that show the development work that you've done on the paper, there's little harm in showing it to people that you trust. If you don't trust your former supervisor, then I'd wait until you've submitted it or even have an acceptance in hand. 

That being said, I'm wondering why you don't already know whether they should be a co-author and haven't already discussed the fact that you're writing a paper on which they might need to contribute. You should at least discuss this with them if there's any contribution of theirs in what you're writing up.