Apologies if this has already been covered.
I am using Adobe Acrobat Pro DC and want to run a script on the digital signature field such that when the user clicks on the field, the application alerts them of any required fields that were left blank and won't let them sign/save until said required fields are filled in (text and radio buttons alike).
You can't do that in the field itself. What you can do is hide the signature field until all the required fields are filled-in, and then show it so the user can sign it.
As for saving the file, you also can't prevent that. What can be done is to display an error message or save a blank version of the file (but that's a bit extreme).