I'd say that that is your main problem here!

For Word to be able to recognise the various parts: First Name, Last Name, Address - Line 1, etc. This needs to be setup in a way that Word can understand it all. But, having them set out as address lables, isn't going to do that, as the details could be different - different address lengths (4 lines for one, 2 lines for another), multiple names, etc.

To really sort this out, you need to have the list in a form where you can have headers to let Word know what each piece of information is. In Word 2003+, you can create the database of names in Word itself, but my choice is always Excel or Access - Excel for a simple, sortable list; Access for a detailed one.

To say that you can't do it from a Label format isn't quite true, but let me give you an idea of what you would need to do in VBA:

First to reverse concatenate the name, ensuring that Mr/Mrs/Ms/Miss/Dr/etc. are taken into account.
Next Break the address down into the various segments - first devising a way to have shorter and longer addresses recognised correctly (definitely NOT easy).
All of this would have to be written in VBA, as there is no way to do it using the Wizard.!!!

As you can see, it would take A LOT less time just to rewrite/export the list to excel! (try importing it as a CSV, you might find it easier to play around with it that way.