Business entity for US citizen living in Australia
To dissolve a Corporation you need to file Articles of Dissolution with the Secretary of State of formation. Depending on the state you might be required to first file a final tax return with appropriate State Department, obtain consents from several departments, etc.
A sole proprietorship is basically yourself. This is not an entity separate from its owner, so you don't need to register it as such with any authority (a DBA is registered as an alias, not a separate legal entity).
Since you are a US citizen technically you could continue your business in the US as a sole proprietor. You might want to register a DBA for yourself in the state and county where you maintain your US address (usually you need to have an address in the US associated with this DBA). Keep in mind though that to register a DBA for a sole proprietor in most of cases you would need to notarize the DBA application, which could be a problem if you are in Australia.