Did you turn on two factor when you changed your password? Also, change the password for the email associated with your account and turn two factor on for that too if you haven't already.
If it makes you do 20 of the maze mice captchas, just back out or close the tab and open a new one to try to sign in again. That refreshed it for me and then it only made me do 3 of them.
Somehow...I don't know how...my Target order went through twice. Even though it kept giving me errors and never actually gave me any confirmation while I was on the website. But then I checked my email and I have two separate confirmation emails with different order numbers for pickup tomorrow...
Have you actually done this? What happens to the old Switch? Is it still usable or does Nintendo fuck with it somehow? What if you don't provide the serial number of the old Switch? Could they just download your AC backup from the cloud regardless? So many questions lol