Hi CCallbaugh!
I also had this problem for a long time and have found the way to fix it. Actually there are many correct posts, on this subject, but some people (like me) just can't get it right the first time. The entry box you need is already on your screen and your Appleid password is already showing in there as *********. OK?
Here's what you have to do:
DON'T just enter the verification code in the password box. You MUST first enter the Appleid password (again) and THEN immediately follow with the verification code.
I know that the password you had entered looks like it is still in the box as *********, but it will disappear when you enter the first character of your (repeated) Appleid password.
Hope this helps.