Is there a way to insert a formula or reference a cell on another
worksheet in the middle of a text string.

For example if I wanted to say: XYZ company will pay $___________ of
your shipping costs... Where the blank was a reference from another
sheet.

One way is to use a formula like this:
="XYZ company will pay "&TEXT(Sheet2!A1,"$0.00")&" of your shipping costs"

The ampersand does concatenation of strings.