There are two ways to put a graphic into a document: floating and inline.
Inline graphics are part of the paragraph text, and do not have anchors.
Not sure what's going on with your anchors that snap back to the picture.
'Move object with text' affects what happens if the anchor paragraph moves:
if checked, the graphic moves when the anchor paragraph moves (ie, the
graphic is positioned relative to the anchor); if not, the graphic remains
stationary even when the anchor moves (ie, positioned relative to the page).