I'm not going to go through the whole story, but my girlfriend is 39 weeks and 2 days into her pregnancy now. Things are confusing, it's either my baby, or another guys.. We are going to try to get a paternity test to sort that out, but regardless of who's baby it is, I want to sign the birth certificate, I love my girlfriend, I'll be proposing to her soon, and I love the baby like my own, even if there's a possibility she isn't... But the thing is, I want to know if I can sign the birth certificate, I want to so much, my girlfriend wants me to sign it too.. Also, we live in Arizona..

Then take her to court as soon as the baby is born and get a court ordered DNA test first. If your not the bio dad and since your not married then you shouldn't be signing the birth certificate. Do this the legal way as you never know what's going to happen down the road. If she nor you are not sure who's it is then you have some really hard thinking to do because this is just a glimpse of the future for the next one.

This is a borderline issue. According to AZ law it is illegal to falsify information on a birth certificate. However, as I read it, you can complete an acknowledgment of paternity form if you and the mother agree and you both believe you are the father.
https://www.azdes.gov/InternetFiles/Pamphlets/pdf/CSE-1043APAM-PD_english.pdf

What I'm not clear on here is whether you can complete the form if you are in doubt. I think you need to propose to your girlfriend and agree whether you to submit the form. It is still possible for the other potential father father to challenge your paternity even if you submit the form.

It might be a better tactic to wait until the child is born, get a test and, if you are not the bio father try to adopt once you are married.

In another question I asked, someone answered with a link to a .pdf file CSE-1043APAM-PD (https://www.azdes.gov/InternetFiles/Pamphlets/pdf/CSE-1043APAM-PD_english.pdf ) . Basically, my girlfriend's pregnant, there's a possibility of the baby not biologically mine. Both of us want me to have legal rights to the baby, so I need some help in understanding a line from that file.

If the mother is not married at the time of birth, a judge can declare a
Man the legal father of a child after judicial proceedings. (Page 2)

Now, does this mean I can have legal rights of the baby? Could somebody please help explain?

I think if you sign the birth certificate you could very well be committing fraud; also, at some point down the road the biological father could show up. Have DNA testing done, then decide how to proceed.