She actually did not run as an independent - the law there allowed her to run as a write-in and keep the Republican Party affiliation - even listed as such on a list of write-in candidates as Republican.
This time around there will be no party primaries - so these articles who keep posting that are apparently not aware of the rule changes. The top 4 finishers in a jungle primary will advance on to the general election - as the incumbent Senator she will most likely make the general election ballot.
Thanks for this information, I was not aware of the change to the Alaska primaries.