Authentication (E-mail) worry
If you click on “My account” and the bring up security you should see the email verification listed. You could download an app for your desktop to use as an authenticator but that would cause issues since you said you play on different computers
Email authentication is still a thing, however ArenaNet has decided that it is not enough. It is good enough if you know what you’re doing, but the problem is, most people don’t, so email authentication is pointless when they additionally get their email hacked (but then they have much bigger problems to worry about) or casually accept the IP verification request.
The alternative to SMS is using an authenticator. Any device that can display a web page can run an authenticator. For Windows, WinAuth is recommended. Unlike SMS, you can setup your authenticator on as many devices as you want by simply using the authenticator’s secret key that you’re given during setup. That is also its downside however, since the authenticator is useless if that key is ever compromised. If you have a USB drive, simply stick your authenticator on there. If not, simply write down your secret key on a piece of paper and carry it with you whenever you’re planning to use it somewhere else.