Which posting date to use in such cases should be discussed internally with Finance.
In my company backposting are only allowed a few days at the beginning of the new month until Finance reporting has to be completed. Month closure would be meaningless if people can still post in the old month after the closure of the books.
As you have a delivery note in your document flow it can't be "not yet processed" in my opinion, but you should check with your SD consultant in detail. Also read OSS notes, they often give background and design information. The correction reports are mentioned or even delivered with the OSS notes.
here is document that lists a few: Useful Standard report to check SD Customizing and Transaction discrepancies
I would definitely search for OSS notes to get more information about the reports, the warnings for execution are often not mentioned in SCN. Sometimes it will even harm your system if you do not understand the processes and just execute such reports. e.g. after archiving, because then the delivery might be archived and the recalculation of a sales order status would reset the status and you can create new deliveries.
Direct update is not an option for me. This is often a considered option from people who do not know how integrated SAP is, and people who know SAP in depth don't consider this because they aware that they might miss an important part.
If you are doing archiving then I bet you will stumble over some cases that can be traced back to direct table updates. That is where my experience comes from.