When you use the "Add Text" method, you are not adding form fields, you are adding real PDF text. This means that when you print "Form fields only", that text is considered part of your document and is not printed.
The good news is that you can actually do what you want to do, but it's a bit more involved. Open a scanned version of your form, then click on the "Prepare Form" button on the right hand pane. This will ask you about which file to use, just double-click on the one that is open (if you have not yet scanned the form, you can also create a form from a scan here).
This will open up the form field editor. Acrobat may have recognized some of our form fields (and it may have recognized areas that are not fields), so you will have to manually edit/add fields. Once you are done, just save the complete form as your new template document. When you open it, you can now fill in the form fields, and these fields will print when you select "Fields only".
