I am new to using Target Premium (at.js). What I'm trying to find out is if it's possible to edit something when on a test page (Cookie, Storage) that would change the variation of the test I'm seeing.
If I land in a test and see Variation C, where is this set? Can I edit something that would allow me to see variation B?
There isn't a "variation" cookie or variable stored client-side to do something like this. Target manages this stuff server-side. If you are looking to QA a test experience or force yourself in to a given version you can use the QA links. Essentially, you can copy a link to the test page with a couple of Target related query parameters that will force you in a particular experience to view it and QA. There are details on this help page: Activity QA.
You can alternatively "activate" a test and then delete your mbox cookie for the tested site, and refresh until you get into the variation you desire. This works fine for AB tests that are randomly allocating the experiences, but if it's an experience targeted test you'll need to ensure you qualify for the applicable audience too.