We need to know a lot more:

https://makemusic.zendesk.com/hc/en-us/community/posts/215729367-General-Discussion-House-Rules 

What version of Finale? What OS? How are you saving to PDF? 

These questions are really important—each one makes a difference. Once we know the answers, the correct solution will be easy. You can do what you need.

Finale 27.3.0.137. Windows 10. From Finale, File-Export-PDF.

That very PDF export may not be the best way to export PDF (as you have found out).

Instead of using

File > Export > PDF

try “Print” to a PDF file.

You can e. g. try CutePDF.

What Peter says. Windows does not have a native PDF driver and requires a 3rd party app. Microsoft's Print to pdf, is one example but that can be problematic.

CUTE PDF is very popular and free—many here use it. You can set up your paper size, print margins etc. to get exactly what you need.

Mac users have different options open to them—again, the correct solution depends on the version of Finale.

Since PDF is (should be) vector driven rather than bit-mapped, the question is whether the new dimensions are in the same ratio as what you asked for. After that, the print size should be what you specify on your printer, and the PDF should adapt to the screen size on your computer.

The above screenshot is after using CutePDF with the A5 size.  Page size should read 5.82x8.26, but it doesn't?

Before creating the pdf, make sure the paper size is correct in File -> Printer page setup, in the Print window, as well as in the Page Format window.

I am not seeing the post here that came in to my email about making sure the print size is the same in all 3 areas.  I made them all the "A5" size, but the screenshot above is still what the pdf size is when I close out of Finale and just open the pdf file??

What are the dimensions of the screen you are opening it on?