Did you read the page I linked?
It tells you how to do what you want to do.
When you say "in the old days, you could freely tab through the blanks on the document," you are referring to a legacy form using the old forms toolbar. (Word 2003 and earlier) I recommend against those controls and methods. You should be using
content controls. When you properly restrict editing or group the controls (as set forth in the
page I linked) you can tab from one to another.
If you use
Rich Text controls, they can contain a tab, so you cannot simply tab from one to another. If you use
Plain Text controls, (the second Aa) they will not have that problem.
Here is a sampler. See for yourself.
If in a plain text content control, tab will take you to the next control. If in a rich text control, tab will insert a tab character.