Hi Matt,
I’ve already seen this problem.

Let’s try to save to IDML and reopen the original document before copy and paste.

This could do the magic if I understood well:

CA$

meaning:
= word boundary
CA = literal CA
$ = end of paragraph

Let’s try it

Maybe Data Merge could help, here you are a good article on this:
https://creativepro.com/data-merge-into-inline-anchored-objects-so-they-flow-in-a-story.php

Jackie, check out the option highlighted in this screenshot, that are the “align to baseline” options in the control panel when you select the text.
In the screenshot selected text is not aligned.
https://adobe.ly/2BP5KUz

Hi Melise,
I think that the Frame Fitting Options are not active by default in the object style, did you turn to active the little checkbox corresponding to the Frame Fitting Options?

Hi Sabrina, I suppose you are writing from Italy.
Data Merge could help you, here you are a tutorial in italian: