Why are you using absolute links? If you were using relative links, of the form */node/22*, the problem would not arise.

If the links are in the node body, and if you have direct access to your database, you can update the text by running a query on the `field_data_body` and `field_revision_body` tables.

First, back up your database in case this goes wrong.

Then run the following query on your tables:

    UPDATE field_data_body SET body_value = REPLACE( body_value, 'a href="www.mysite.com/','a href="mysite.net/');

Repeat with the `field_revision_body` table.

This should change all references to the old domain with the new domain.