It is always possible that Contact Form 7 didn’t install or update correctly on your site. It’s a general problem with WordPress updates due to the unreliability of internet connections.
Delete the plugin folder at wp-content/plugins/contact-form-7/ using FTP (don’t use Delete option on Plugins page as this may erase all CF7 forms) and install the plugin again.
Is there a way to backup the forms so I don’t have recreate them from scratch?
You may backup the form by exporting your form. This could be done from tools >>> export and then select contact from and save the xml to your computer.
You can restore the file to have your form settings any time you need it.