A WordPress site is made up of three main components that you'll need to handle to migrate the site:

1. The web files. On a shared host this will usually be the content of your publichtml directory. On a droplet with a default Apache configuration with modphp these will go in /var/www/html and will need to be owned by the www-data user.

2. Your database. Most shared hosting provides phpMyAdmin to access your database. From there you should export your entire database as a .sql file. This is a MySQL script that includes the commands to create all needed tables along with the data to be imported.

3. Configuration: On the web server configuration side there shouldn't be a lot to do here since the default configuration will work well for WordPress in most instances. You will also want to repoint your DNS either by using DigitalOcean's nameservers or by pointing your domain's main A record to your droplet's new IP address.

As far as SSL goes, while a self signed certificate can be useful for testing or for applications which will run locally for a small group of users it is not recommended for use on a public site. Visitors who view your site will, by default be shown a warning that the browser cannot confirm the authenticity/security of the site. This is because there is no issuing authority to guarantee that the information provided is correct. I would recommend using LetsEncrypt who offer free SSL certificates which are publicly signed and accepted by all major browsers instead.

If you can share some more details about the exact issues you are encountering and at what point in the process you are encountering them it will help us to better help you.

Maybe try https://transfer.website . I use it to transfer websites between servers, works ok so far.