It's over 15 years since I used Word 2000, so only some rough suggestions—pending a response from a more expert poster.

Copy 'My Template.dot' and rename the copy to 'Normal.dot';
If there's somewhere in the template properties to 'automatically update all documents', you should select that.
I recommend you make a backup copy of the folder(s) with your old docs though, so you can reverse anything that goes wrong.

Alternatively, open each doc and find the 'Attach template' function in the menus. Attach My Template.dot to the doc, and select something like 'automatically update styles'. Then save the doc.

PS Normal.dot is around somewhere, when you delete or rename it, Word will automatically recreate it next time Word is opened.