Seeking Advice on WordPress Server Migration for my directadmin

Are you a Directadmin license holder
Because this is a forum for support to Directadmin license holders, so admins and sometimes resellers, but not for end customers using Directadmin.
You should ask support at your current system admin who is admin of the Directadmin installation, it should be no problem at all for him to help you with this.

Just to be of a bit of help (there are lots of movement instructions for Wordpress to be found on internet), if your new server will have Directadmin too, use the admin backup/transfer if you're an admin.
If you're a hosting customer, you can just use the backup option from DA in your user account and let your new host restore it again.
There might be need to change paths in Wordpress.

Also have a look here: