Vertical line Text gap

How can I adjust the spacing between the text and the vertical line when text is inserted into the vertical line?

In Engrave mode, select the line and open Properties (the lower zone).
In the Horizontal Lines section, there are two relevant properties: Text Position (Above, Centered, Below, Inside, Outside) and Text offset (spaces). Fiddle around with those and you will probably be able to do what you want.
There are also properties there for Text placement (Start, Center, End) and Erasure padding (L, R, T, B).

Edit (after re-reading your post):
My comments apply to horizontal lines. I have never tried this with vertical lines, so you might need to experiment a bit.

Thanks.

There is a text offset property for vertical lines in Engrave (if you bothered to look for it).