This may sound like a foolish question, but do I have to be connected to the internet to update with SwareT?

The simple answer is: Yes

The more complicated answer is: How else can you get the updates? Unless you use swaret on another computer, save them yo a CD, and tell swaret to use the CD

To expand on cwwilson721's answer:

Depends how you have set things up.

SWareT is going to look for upgrades somewhere. Most people have it set to look for upgrades on the internet.

Another reasonable solution would be for a computer on your network to be a repository (easily achieved through the use of rsync).

Since SWareT is a tool to update your system, there has to be somewhere to look for updates. Though the Internet is the obvious choice, local repositories can also be used. Any local repositories will have to get their updates from somewhere, though. Thge way I see it, you can use the Internet or a form of Sneakernet.