The Escort the Dignitary one you get from the Operations officer on New Romulus won't complete until the dignitary beams down...
But, the ship flies to a pojnt, turns around and flies back to the transwarp gate without beaming down the dignitary.
I've encountered this 3 times today alone and have tried flying in front, behind..no joy.

And BTW, the Inspect Cadets one on the StarBases STILL gives cadets wioth no corrections about half the time.

*bump*

Both OP's issues still occur

Holy cr@p! Is this dang mission STILL bugged?

The counter is still at 0:00, and the dignitary does not seem to have beamed down.

I transport to the Embassy, won't let me turn in the mission to the Officer of the Watch. I transport to the Staging Area, it still says Go To New Romulus System in the mission tracker area of my HUD.

Fix it Cryptic!!