Dear Teachers,

In the following message would it be mor appropriate if i use sorting out
instead of settling :

With Reference to your messages regarding a/m subject,
well noted with thanks.

Please note that subject vessel sailed from our ports
on 30th June (noon time) after settling all the claims
against the vessel.

The question being asked is the difference in meaning between "settling" a claim and "sorting out" a claim.

Both could be taken to mean the same thing. If a claim is "settled", this implies that money (or some form of compensation) was paid out. If a claim is "sorted out", this can also mean that compensation was paid. But another interpretation would be that compensation wasn't paid, and instead the claim was successfully disputed. In other words, "sorted out" is not specific as to how a claim was handled, it simply means that it was dealt with in some fashion.

So, if you want to emphasize that all the money that was due has been paid, you should write that the vessel sailed after settling all the claims against it.
taiwandave covered the initial question well, I don't have anything to add on that.

But if you want to talk about date and time you should be saying June 30th or on the 30th of June. Also, if you have time it's generally referring to a time zone, so you'd want to say noon, June 30th or at noon on the 30th of June, or on the 30th of June at noon, or 12PM June 30th (yes, noon is 12PM, I know there is confusion over this).