It sounds like you are using the Text Tool to enter these numbers. That's not the way - use the Expression Tool instead. The Text Tool is only for page related text like title, composer, instrument label, copyright, etc. The Expression Tool is for music related text that will always "travel with the music".

I'd make a separate expression category for these fingerings. In the Category Designer, set the justification to "center", the horizontal positioning to "center of primary notehead" and the vertical positioning to "vertical mouse click". That will at least get them near ideal positioning.

Thanks. Sounds like what I'm looking for.  I'll try to set it up the way you suggest and see if I stub my toe.

Bruce's suggestion is a good one. But my additions to the Expression tool only stay in the one specific score.  Can I get Finale to save my changes/additions to the Expression tool for use in other scores?

You'll have to save those as a library.

File>Save Library>

Depending what you have created and what you want saved, you should select one or the other, or both. Please note though that because of the way Finale is it saves all text/shape expressions, so if you have something in any of the default categories, those will be saved along with your custom category. The way to get only yours is to delete everything except what you want to save. An easy way to do this is to make a copy of your file and go from there.

In the future if you want to create an expression library from scratch, it might to useful to start a new file without the default libraries loaded. To do this go to:

File>New>Document Without Libraries

Once saved, you can then load your expression library in any file you want.

That did the trick.

What I'd done was just add fret numbers (Roman) and finger numbers (Arabic) to the Miscellaneous category. So after saving I now the default expression categories that include these for use on any score.

Thanks!