However the format it outputs the custom field is just a raw date. How can I format the output of this custom field to be more pleasing? Add format=‘F j, Y’ doesn’t work.
That will not work because the custom_field shortcode literally just outputs the contents of the field, it could be anything, text, numbers, a url or an image, anything.
So you need to write your own shortcode for wordpress. Twenty odd years of documentation on the internet for add_shortcode Heres an example: