Since this is not a request for submission status, it would fit better in the "Submissions" forum.
However, to answer your question, go to each dmoz.org category where your site is listed, anc click on "update URL" at the top of the page, and follow the instructions.
It will help if there is some indication on the old site of the shift to your new site. A 301 permanent redirect is the best way, but others will work. If there is no indication, and both sites work, an editor will most likely not approve your update request -- this policy helps to protect both the ODP and website owners from hijacks.
