There is no need to publish any private posts in order to do what you want.
Let us have another look at your request though.
Is there a way that they can allow a certain user to write the posts, but not post them (at least not in a way visible to the public)?
There are two parts to this question.
...allow a certain user to write the posts, but not post them...
If you mean that the user will be able to write a post but at the same time, systematically by Facebook, not allowed to publish the post, then there is no simple way to do what you ask for (maybe not even a way).
...(at least not in a way visible to the public)
There is a very simple way to do this.
It involves at least two parties working together as a "team".
- Someone with the role of
Admin
for the page acting as the "Supervisor" that will review and decide whether to approve the posts and publish them
- Someone assigned the role of
Editor
that will be the "Creator" of the posts.
The "Creator" will write the posts, save them as Draft
and then the "Supervisor" will, at a later time, review them and decide for whatever action needs to be taken.
The steps:
From your writings I assume that your organization created the Facebook page, which makes them the "Owner" as well as the Admin
.
Then, you were later assigned as the PAO granting you permission to write the posts that go online. This makes you an Admin
as well or an Editor
(depending on the "Page Role" you were granted for the specific page).
Step 1: Create a draft of a post for the Facebook Page:
- At the top of your Page's Timeline, click the type of post you want to create
- Add details to your post
- Click the little arrow next to
Publish
- Select
Save Draft

You will get a post on your page informing you of the created draft.

Please note: If you have more than one drafts they will be "bundled" together to just one draft post.

Your job as the "Creator" of the post is done.
Step 2: The Admin of the page wishes to review/publish the post:
- The
Admin
logs-in under the organization's account.
- To see your drafts, will then need to click on
Publishing Tools
at the top of the Page,
- Click
Drafts
in the left column
Or simply click the See drafts
link on the above mentioned post.
This will bring up the Drafts screen:

The Admin
can click on any draft and review it.
After reviewing the post/s can now:
Edit
or
Publish
(approve) the draft/s you've created,
Schedule
,
Backdate
or simply
Delete
it/them by clicking the appropriate link.

Draft posts can be edited and re-edited, saved and re-saved as many times as you like.
Following the above steps does not require any special coding or actions. Just an understanding between the involved parties of the organization.
As mentioned before I assume there are at least two unique accounts for the page.
If -for some reason- the structure of your organization is different or you need to edit a role or even assign a new one for your page (you may want to have more than one Editor
for when one is on leave) you should use the "Page Role" screen under Settings
:

Here is a table you could consult about each Role's permissions:

References at the Facebook Help Center:
How do I create a draft of a post for my Page?
How do I edit or publish drafts for my Page?
What are the different Page roles and what can they do?
How do I give someone a role on my Page?