Can I receive both an NSF international postdoctoral award and salary from a foreign university? Recently I was offered a full salaried postdoctoral position (not a fellowship) in Australia (with a long lead time).

I am a US citizen, so can also apply for the NSF international postdoctoral fellowships.

Suppose, hypothetically, that I apply for and receive an NSF award.

Is it acceptable to increase my postdoctoral salary with the fellowship, or would it not make a difference?

Given that I already have an offer in hand, if the NSF fellowship wouldn't increase my salary, then it isn't worth it to me to apply (I realize there are other benefits, prestige, that go along with receiving a fellowship, but I'm time-strapped at the moment).
There will be rules that NSF sets for these fellowships. You need to look them up and/or talk to the program officer who administers the grant.

My best guess is that you can't double dip. So you will likely have to decline one salary or the other. On the other hand, there is nothing wrong with still listing the grant on your CV -- it is a grant you won in a competitive solicitation, so it belongs on your CV.