Can you migrate MySQL data bases without access to an Admin Panel using FTP

[Imacyn]

This is a question from a lady that knows very little about the subject. Please someone answer to help me save my model hobby photo program.

The current server my files are on, the ISP no longer supports a PHP MySQL MyAdmin Panel. It stopped working a couple of days ago. My ISP wants to set me up on new server with my own blade and Plesk Panel. Problem is I have to do the migration myself and can not find the php databases in my WS FTP window. (Not sure if they would even be there is the first place) Can you perform a successful migration of all php databases without an access to the Admin Panel? I have a total of three sites and not a lot of money. I thought I'd try moving the smallest site first to see it the databases or PHP script has to be re written to perform the program I am using which is a custom program. We make no money from this and it is a hobby about 500 mostly ladies have fun taking photos of model horses, dogs, cattle and cats and compete in a judging format. Since I can't access the admin panel I can't make repairs to the show results and so our hobby is doomed without some expert help. Thank you in advance,

[mwasif]

You can take the backup of MySQL database if you have shell access using mysqldump.