As a first step I'd make sure with the journal that they haven't just failed to update the author info page - my experience with those is that they tend to be in terrible shape.
Using outdated APA is really a slap in the face of authors, partly because all the good online citation guides are for the most recent editions.

Zotero had a pre-6th edition APA style, that can probably be dug up from the archives, but if I remember correctly, I also fixed a whole bunch of general issues with APA in the process of implementing 6th edition changes, so I'm not sure how much fun you're going to have with that.

look here
http://ocls.cmich.edu/APA%206th%20Edition%20CHANGES.pdf
or here
http://flash1r.apa.org/apastyle/whatsnew/index.htm
it's actually pretty minimal - there are only two major changes for references-
the inclusion of the doi in all cases in 6th ed. and the different listing of multiple authors.
The former is pretty easy to change - I can see if I get to that sometime this week - the latter is not implemented anyway, because Zotero can't do the first authors ... last author thing that APA now requires for more than 6 authors - anyway, that one probably won't affect you in polisci anyway.
Please do check with the editors, though, about changing to 6th edition.

Zotero automatically gets the doi for a lot of journals - and as an increasing amount of styles want it, you should keep it/add it - it's really no big deal to change the style, I'd do that.

Read the first link, and use the second as a reference.
csl_simple_edits [Zotero Documentation]
dev:csl_syntax_summary [Zotero Documentation]
You will just need to remove the part that adds the DOI to each item in the bibliography section of the style.

At my Faculty I am also required to use APA 5.

I have just added an APA 5th addition to the repository - I haven't tested it thoroughly - please report back with any problems.