I am sorry to hear this! Is the button you are referring to the "Add Another Phone" link or the grey, rectangular box where you can choose which type of phone number it is?
Are you trying to add a contact while on a phone or tablet? If yes, our site is not optimized for use on mobile browsers and I have seen that features like dropdown menus might not perform correctly.
If you are on a desktop I recommend trying the site in another browser or in incognito/safe mode. It's possible there's an extension or plugin that's interfering with the site experience.
If that does not help, reach out to us at social_support(at)constantcontact(dot)com with your username and a reference to this post and we can investigate further!