As others have said, replace the URLs in the database. If you're comfortable with the command line, and your host allows it, install wp-cli and use it's search-replace tool (i.e. wp search-replace olddomain newdomain).
Otherwise, I recommend this search replace script:
https://github.com/interconnectit/Search-Replace-DB . It will replace the domains in every single location of your site, including serialized data. I've been using it for years with zero issues, apart from typos :)
Regards,
Ben