I've successfully added one widget to my Story Map but I cannot add a two Twitter widgets and a weather widget.

Owen,
Thank you for your help thus far, I've figured out some work arounds for the meantime.
I will be working with an Esri Rep sometime next week to go over some of these issues.
Thanks again!
I was wondering if you have worked out this issue because I am also having the same problem:
Embed code can only contain one "<iframe>"