I recently switched to a new phone and downloaded the security key app. Still having the old phone and old app, I went into it, got the serial code, and tried to input it into the app on the new phone which asks for the serial code. It is not accepting the serial code however, and is instead saying the code has an incorrect number of characters. Can anyone please tell me what I'm supposed to do?

You need to remove the old Token first, simply log in without the Code, Click on the link for lost token and then proceed.

As you still have the old phone, use it to log in normally to the web site, then go to the account management section of My SWTOR and remove the old app (you'll need to give a one-time password that you get in email). Then add the new one as if it is your first time.

 

(Just went through this myself, when iTunes didn't restore enough information for my iPhone 7 to be able to use the SK app that had been on my 5S.)