I had the same problem - luckily the solution (found here) is very simple!

1. Insert the reference as you normally would
2. Right click the reference (the gray field)

3. Select "Toggle field codes"

   You should see something like this:

   { REF _Ref218925266 \h }
   

4. Simply change this to:

   { REF _Ref218925266 \# 0 \h }
   

Voila! Now Word persistently only shows the number.

(Right click to "Toggle field codes" once again, to get back to normal text display.)

PS Don't forget to "update field" otherwise you might think that it didn't work!

Almost, the following will do what you want except that you'll also need one body style per level (the same as you need one heading style per level), instead of one style overall. And please note that I'm doing these in Word 2003 so they actual motions may be slightly different.

Build Styles

Get up the styles pane, IIRC in Word 2007 you hit the small down arrow on the bottom-right of the styles selector on the ribbon. Click on the New Style button, on Word 2007 I think this is a pictoral button of a letter A with a gleam/star on it.

Each new style should be Based On a corresponding Heading entry, this ensures they will stay within the correct numbering pattern, because they're using exactly the same pattern. Obviously, you'll need to remove bold, adjust the font size, adjust the paragraph spacing, and whatever else you need to ensure the body text appears how you want it to.

You probably will want to create BodyText1 based on Heading2, BodyText2 based on Heading3, and so on for however many levels you will need.

Instead, you might be able to base BodyText2 (etc) on BodyText1, and just adjust the numbering level, but for some reason this caused me a problem. Not sure why, but you may wish to try so you don't have to redefine the font size/etc for each BodyText style.

Adjust Table of Contents

Because the new styles are based on the Heading ones they will appear in the table of contents. Obviously, this is not desirable.

This is where my memory of Word 2007 memory becomes hazy, the following is purely Word 2003 - if it doesn't match, let me know and I'll update for Word 2007 once I'm home and can check directly.

Right click on your current Table of Contents and select Edit field..., then click the Table of Contents button to get to the TOC window. (You can probably get here when creating a brand new TOC as well)

Click the Options button (bottom-right) and then find each BodyText style in the list; clear the TOC Level field from each of these. This ensures the BodyText items will not appear in the TOC at all. Close each window until you get back to your document, you might be asked if you want to replace the current table, if so click Yes.

Use the styles

Now just ensure you set the relevant style on each paragraph as you write.