http://word.tips.net/Pages/T001743_Pictures_Move_on_their_Own.html
http://www.faultwire.com/solutions-application/Microsoft-Office-Word-*3945.html
You cannot put drawing objects into a text box,…
These were the most helpful links I could find, but none quite exactly addressed my problem. However, after an hour or so of tinkering I figured it out:
SOLUTION: Format the AutoShape as “In line with text” — then you can move cut text from your document and paste it into the textbox under AutoShape with the object (textbox or AutoShape) deleting itself! 🙂