I assume I'm not using the correct part of the date object array- what should I use? The timestamp? Just $_POST['wpcf-from'] ? Something else? I've tried several options and can't seem to find one that works.
During troubleshooting, I noticed that the function that you're using is correct, except for the following line:
$post_id = get_the_ID();
This line incorrectly replaces the newly entered post's ID, with the ID of the post/page, where the form is placed. As a result, the function "update_post_meta" updates the custom fields on the post/page with the form, instead of the newly added post.
Please remove that line and then check the form submission again.