Can prior service wear awards from their previous branch on their Army service uniform? What literature would have the answer?

Check AR 670-1 also.

Going to go out on a MOSTLY certain limb here and say "yes".

My sergeant and I (army) received Navy Achievement Medals while we were overseas, and we were both permitted to wear our navy ribbons on our racks.

Yes. DA Pam 670-1, para. 20-5 provides order of precedence.

I'm prior service Navy and I can wear all awards and decorations from the US Navy except for marksmanship awards.

I’m assuming it’s the same way for all branches of the military. I was in the ARMY from 89-98, got out and joined the Air Force in 2014 and I am allowed to wear all of my ARMY ribbons on my Air Force uniform. They go in order of precedence.

There is a page on HRCs website that describes what you need to do. Google HRC Sister Service awards

I wore my army good conduct medal/ribbon while in the navy and made sure it was right before I did it. Probably works both ways.

I served on AD in the Navy and received my good conduct award. I then went into the guard - first Army then Air. While in the ANG - I wore my Navy Good Conduct award with my Air Force awards.

Years ago, we had a prior-service sailor who earned the Submarine Warfare Insignia (Dolphins) and wore it proudly on his uniform. This gave our former Special Forces first sergeant a conniption fit (which I dearly loved).

Yes, with exceptions. I went from USMC to Army. All previous awards were authorized. But I had to suffer the slings and arrows from the Marines for leaving and the Army for being dumb enough to be a Marine.