There is no option to skip that step.
You would be prompted to save the document once you apply digital signature as now it is a certified file.
Also if you make changes to the document by adding another signature you would need to save the changes.
What's the point of not saving the file after you signed it? The signature is embedded in the document. If you do not save it, it is the same as if you didn't sign it. The actual process is more complicated than that but you need to ask yourself this question, You can save signed PDF in the file with the same name at the same location so that the signed PDF will overwrite the original one, but this is a bad practice, which I never recommend.
Your problem is that saving is a part of signing. First the file is written, then all sorts of signature magic is calculated from the file, then the signature is added. So it has to be saved SOMEWHERE, and while it might be possible to save it silently behind the scenes, then sign, Acrobat doesn't do that.
And by the way, if ONE person has to "sign in different places" this is fundamentally wrong - a common error, but completely contradicting the absolute principle that a signature is on a FILE, not a page. This encourages bad practice, i.e. looking at pages for signature appearances, which should never be done - I wish they had been all invisible.