There is a macro-free way to accomplish this, though it is not very intuitive. When you click in the Combo Box Control, the UI for the box appears, with the drop-down arrow on the right and a vertical ellipsis
... to either the left or top of the box.
Hold down the right mouse button on the ellipsis and drag the cursor to the area where you want the variable to be. Release the mouse button, and a menu should appear saying "
Move Here, Copy Here, Link Here ...". Choose
Link Here and Word will create a link in that spot to the value of the combo box, and the link will auto-update whenever the combo box changes.
Downside: this solution may cause Word to display a "potential external link warning" when opening the document.