We offer free WordPress or HTML website transfers / migrations to mywebhostingaccount.com. You can contact us and ask about your specific situation.
If you already have a web hosting account with us, you can use the "Website Transfer" contact form ticket from within your account, to submit a request for a website transfer / migration. In the website transfer ticket, there will be a place to put in your old web hosting control panel or WordPress admin login information, and a place to let us know of any custom settings for email (MX Records, email accounts, or email forwarding) and/or DNS records (redirects, subdomains, TXT records, etc.). With the needed information to login to your old website hosting control panel to retrieve the files and settings, we can attempt to transfer your website to us. The process will not involve deleting the website from the old host, but just copying it over to us.
If you are a DIY'er, there are some tools you can use. First there is the WordPress plugin "Migrate Guru", which is a free plugin that enables you to migrate any size website up to 200 GB, by handling the transferring on its own servers. This plugin can be installed from your WordPress admin area, in the "Plugins" section, by clicking on "Add New", then searching for "Migrate Guru".
Another tool you can use to transfer your WordPress website is the "All-in-One WP Migration" WordPress plugin. This plugin can create website export files that you can download to your computer. You can also use this plugin to create WordPress backup files that are stored on your web hosting account file system. To transfer your WordPress website using this, install the plugin and export the website file from your old WordPress admin area, then install the plugin and import the website file from the new WordPress admin area. You will need to create a blank WordPress site at the new web hosting account first in order to import your website using this plugin. This plugin is free and available to install by searching for it in your WordPress admin area plugins section, but in the main free version that is available, there are limitations to it, such as website size limits.
You can get this "All-in-One WP Migration" WordPress plugin for free without as many limitations, from a modified version, available to download as a .zip file on GitHub here: github.com/d0n601/All-In-One-WP-Migration-With-Import
You can also download this modified version of the "All-in-One WP Migration" WordPress plugin with less limitations, as a .zip file when logged in to your account on mywebhostingaccount.com, from the "Support - Downloads" page.
For manually installing a WordPress plugin via a downloaded .zip file, you can refer to this outside article: Install a Plugin using the WordPress Admin Plugin Upload