The posts in WordPress are stored in a MySQL database. There are several ways to access a MySQL database, but the most common way is that your web host provides access to a tool called PHPMyAdmin. This is a web application that allows you to manage your MySQL database, including making an SQL dump. An SQL dump is a file with the data of your MySQL database. It is possible to import the SQL dump in your new database.
Basically, when you install WordPress on your new domain, you should be able to reset your new database and import your SQL dump again.
For information on how to do this, contact your web host – some use other software than PHPMyAdmin.
Good luck!