Solved! Go to Solution.
Update: That was a frustrating day. My wife & I spoke with customer service on the phone. They told us the FTP Username/Password were "being updated but are backed up" and that it "may take around 3 hours" to work properly. I ended up waiting 14+ hours and the FTP username edit page still says "Pending Update". Eventually I created a new username & password which began working in about 30 seconds.
Something is broken on GoDaddy's side, but at least the new username & password work-around "fixed it" well enough that I can use my server. I hope this helps someone who searches for this problem one day! :-)
Also note that there appears to be an additional 'secret' security requirement for passwords which is not listed as a requirement: the password cannot contain more than 2 repeated characters next to each other.
I had repeat numbers (111) as part of a sequence, and on a retry I had repeat exclamation points (!!!), both of which the godaddy password reset accepted as valid, but which failed when actually using them in FTP.
Changing the password to remove repeated values allowed FTP to accept the credentials and work normally.
Allowing repeated characters increases the variety of possible passwords, but it also encourages people to use easily-guessed passwords like 'a1111111'.
Not sure if this is actually a thing, but it's worth trying if you've changed your password and it has 3+ of the same characters next to each other. Worked for me.