Welcome to the forum!
There are a few ways to make those edges meet perfectly. And of course, the way they are now, that would be impossible, because one edge has curves rather than the sharp corners on the other one. So one of the sides will need to be changed, in addition to being "moved", essentially.
Having the stroke only on the outside or inside is a feature which has been requested for as long as I've been using Inkscape. I'm starting to wonder whether an official request has ever been made for it. But anyway, the only way to do that currently, means that the stroke will be a separate object. And considering the whole image, it's the strokes rather than the fills which will be touching, in the area where the gap is closed.
I'm curious. In the area where the gap is closed, and the 2 stroke objects will meet, is it required that there are actually 2 strokes there? Is it required that there is a top piece and a bottom piece? I guess I can explain myself better with some examples. Let me make a couple of quick drawings, and I'll post some screenshots and explain.