Well as an archaeologist I can tell you that an excavation by it's definition is always destructable. Once you excavate something, you can't undo the procedure. That's why we have to be very careful in our documentation, because every information we missed is lost forever.

What you want to do is get into field school somewhere, preferably under a director with a great reputation. With that under your belt, you can find paid internships at various dig sites. You'll end up needing scuba training and a certification as an AAUS scientific diver if you want to work in nautical archaeology.

I've only worked on dry land but a buddy from field school went on to get more training, experience, and certifications. Nowadays he lives and works in Malta and gets to dive ancient shipwrecks in the Med. The lucky bastard.