Oh, and when I add the first note of the next measure, Finale adds a 32nd rest, and will allow insertion of a 32nd, but not a 16th. There is just a dotted half-note, and then this septuplet 16th run, in the bar. Thanks!

A search for tuplets shows that you are not the first to face this kind of problem, nor is there any lack of solutions to your situation.

https://makemusic.zendesk.com/hc/en-us/search?utf8=%E2%9C%93&query=tuplets 

Thanks Adrian, I have been reading some of these entries, and did check out Mike Rosen's tuplet tutorial and tried following his steps. I got the idea from some of these to try entering this on not the final beat of the bar, and after some messing around, got it to work. Thanks!

Often, even following my steps, there will be a superfluous rest at the end of tuplets beyond a simple triplet. If so, just delete them and move on!

For sure, thanks Mike! The problem today for me was that Finale just wouldn't let me add in the 7th 16th, despite showing a "7" over what it seemed to think should have been a grouping of 6 notes...I was very happy to find a work-around!