EarthLink is a webmail based client email program. You use their server to retrieve and send emails through your browser. When you do not use your browser to connect, you cannot retrieve any messages.
Webmail requires an internet connection to work; if your internet connection is down, then so is your email account as it requires a connection to a web server to work.
You must login through your browser.
What this means is, your computer cannot send off spammy messages without your knowledge as you must always first connect to EarthLink to get or send your mail.
Programs such as Microsoft Outlook, Mozilla Thunderbird, IncrediMail, OS X Mail, iOS Mail are all programs installed on your computer or device. These programs access their email server at all times as long as there is a network available. They do not require a browser to run and work.
You must login through your program. You do not use a browser when your email program is installed.
To clarify, GMail is a webmail based email account. You access it through your browser.
Like EarthLink, you can read, send and/or organize messages. The difference is, you can get your email off any internet connected device using webmail, but installed email programs work only on the device they are installed on.
Why is this distinction important? If your device is infected it can send out spammy or spoofed messages if the email account is installed on your system, but cannot if you use a webmail account.
In short, it is up to EarthLink to fix your account.
You can’t fix it for them.
[EDIT:] (Quote taken from the following site: https://www.techwalla.com/articles/what-is-the-difference-between-webmail-and-email)
Webmail can be accessed from any computer, making it convenient when you don't have your computer on-hand. However, entering your authentication information into a public computer can be dangerous and may expose your account to hackers.