UPDATE:
I realized that I had a “browser” key, not a server key. So I got it, and replaced the key in my settings. However, it still doesn’t work. Here’s what I get in my PHP error log when I try to add the videos:
2016/02/08 19:12:42 [error] 6622#0: *2903 FastCGI sent in stderr: "PHP message: WordPress database error Column 'post_content' cannot be null for query INSERT INTOwp_posts(post_author,post_date,post_date_gmt,post_content,post_content_filtered,post_title,post_excerpt,post_status,post_type,comment_status,ping_status,post_password,post_name,to_ping,pinged,post_modified,post_modified_gmt,post_parent,menu_order,post_mime_type,guid`) VALUES (4, ‘2016-02-08 19:12:32’, ‘2016-02-09 00:12:32’, NULL, ”, ”, NULL, ‘inherit’, ‘attachment’, ‘closed’, ‘closed’, ”, ”, ”, ”, ‘2016-02-08 19:12:32’, ‘2016-02-09 00:12:32’, 5235, 1, ”, ”) made by wp_insert_attachment, wp_insert_post
PHP message: WordPress database error Column ‘post_content’ cannot be null for query INSERT INTO wp_posts (post_author, post_date, post_date_gmt, post_content, post_content_filtered, post_title, post_excerpt, post_status, post_type, comment_status, ping_status, post_password, post_name, to_ping, pinged, post_modified, post_modified_gmt, post_parent, menu_order, post_mime_type, guid) VALUES (4, ‘2016-02-08 19:12:35’, ‘2016-02-09 00:12:35’, NULL, ”, ”, NULL, ‘inherit’, ‘attachment’, ‘closed’, ‘closed’, ”, ”, ”, ”, ‘2016-02-08 19:12:35’, ‘2016-02-09 00:12:35’, 5235, 1, ”, ”) made by wp_insert_attachment, wp_insert_post
PHP message: WordPress database error Column ‘post_content’ cannot be null for query INSERT INTO wp_posts (post_author, post_date, post_date_gmt, post_content, post_content_filtered, post_title, post_excerpt, post_status, post_type, comment_status, ping_status, post_password, post_name, to_ping, pinged, post_modified, post_modified_gmt, post_parent, menu_order, post_mime_type, guid) VALUES (4, ‘2016-02-08 19:12:37’, ‘2016-02-09 00:12:37’, NULL, ”, ”, NULL, ‘inherit’, ‘attachment’, ‘closed’, ‘closed’, ”, ”, ”, ”, ‘2016-02-08 19:12:37’, ‘2016-02-09 00:12:37’, 5235, 1, ”, ‘`