Yes, if you start a full import, for the old posts to not be imported, they must already have been imported. It does not matter whether they stay in draft or unpublished state - the plugin just needs to be able to know that they were imported.
If you start a new import, this is what will happen - all posts get synced.
The date is intentional - it reflects the date of the import of the post into WP, not the original publication date.
Because it didn’t import this morning
Syncing of new posts after one run of post import completes is done differently - they are done through webhooks that contact your site and send the post data. If your host or website prevents these connections in any way (security, load on your server etc), then the syncing of new posts dont work.
If you dont want to import old posts, delete all of them and dont start any kind of manual import.
If new posts are not coming in, the site may have lost its webhook - disconnecting the plugin, deleting all plugin data and going through the setup wizard again (including post sync part) should help re-create the webhook.
If you still dont receive new posts after that, a security issue at your site or server can be suspected and must be investigated. I can also look into your account’s webhooks on this side as an added measure.
Open for resolution that doesn’t involve blaming a paying customer of your pro plugins.
The issues i mentioned earlier and in this post are common issues stemming from website setups and host setups. They constitute a significant portion of issues with the plugin, and they cannot be solved without talking about them.