If you insert measure stacks before copying the new music in, that may preserve some of the formatting. Pagination, measure numbers, etc., will update. Try it ON A COPY before committing yourself!

Thanks, that was what I was thinking, but of course the measure numbers will be new.  I was thinking even to format the measures locked into staff systems and pasting by group.  Will definitely try. 

As far as measure numbering goes, you could use Musical Theater numbering for the added measures.

If you put a hard page break on each page it will eliminate some of the reformattting. You will still have to adjust where you added music of course. Keep in mind a page break are different than system locks.

Good idea, put a page break on every page.  Can one do it on several pages at once?  When the measure numbers change, the page breaks stay with the original measures?

Yes, your following and before pages will not change. Of course the page you add measures to will and will need adjusting. But you can decide how you like it even if you have to remove the next pages' break, etc.

There is a plug-in to remove all Page Breaks but I don't know of any that add them all at once. You have to do each page separately.

BTW, if your edits are permanent and you have not released or published your work, I would fix the measure numbers and not use additional letters.

You are suggesting changing measure numbers but keeping rehearsal letters the same.  I could maybe fudge a few of the rehearsal numbers to catch up by C of each scene.  Also a good idea.  The opera is being performed in February and singers and design people are working with the old score.  Everyone will need to put their markings into the new score.  That shouldn't be so terrible.

If it has already been handed out than do use the letters on the additional measures.  Example above, 4A, 4B, 4C, 4 etc. as needed. This is common and standard practice for musicals.

If it has not been published, otherwise not given to the musicians, simply correct the measure numbers.

… Is there anyway to insert the new music into each scene (both the orch score and the parts master) without losing the original formatting of the score and the parts master (which are separate files)? …

To avoid disasters with the system margins, use the Page Layout Tool option {Insert Staff Systems…}.

Do not insert measures.

If you insert measures, and then lock the inserted measures into one system, then all systems after that will be off by one number in the layout (so that e. g. System 42 will have the system margins for System 43).

THANK YOU PETER.  I think this is the true answer.  So I will prepare the music in a separate file and know exactly how many measures I have and how they should be laid out, and then insert staff systems with the layout predetermined and then paste in.  I think that will work.  I wish I'd known this years ago.  and I've been using Finale since the beginning, in the early 90's.  

Finale has many ways to do almost any thing. Make a copy so you are not working on the original. In case one of the many ways that might work didn’t pan out.
If you already have music out and can’t retrieve it, you need to keep as much as possible to the original like using measure numbers with letters.
But work on a copy! Not the original.