We would have an easier time helping out if your could post an image of the situation, perhaps even highlighting what is amiss.

The 3 needs to be vertically centered in the bracket. And despite the setting, the tuplets are appearing on the note side rather than stem, as I specified.

What puzzles me is that, when I create the situation you show from scratch, the tuplet 3's are lower compared to the brackets.

What's puzzling to me is that after I fixed the tuplets, I tried to recreate the problem, and this is what I got:

Engraver tuplets and avoid staff are unchecked. Stem side box is checked. Below, after I checked the  Engraver tuplet box:

Now, after checking Avoid staff:

An improvement, but this was not the result I got when I was trying to fix this earlier. The 3's are still not centered vertically. Despite the downward movement, all but the last tuplets are even. Checking Use Bottom Notes makes no difference.

It's possible to fix this individually (check Allow Horizontal Drag, uncheck Avoid Staff, then move bracket and #), but what a royal pain.

… numbers which have to be manually adjusted to be properly vertically spaced. In order to do this, I have to check "Allow Horizontal Drag" …

What puzzles me is that a setting for Horizontal would affect the Vertical position.

Could you give us an example of that?

I don't think it was Allowing Horizontal Drag that let me move the number vertically, but unchecking Avoid Staff. When Avoid Staff is checked, the number should be allowed to move independantly.The bottom line here is that tuplets aren't handled very well at all by Finale. They need to be overhauled, including adding Above and Below settings as suggested (I think) by Jason Loffredo.