That's what you do in a tense crisis situation. Abandon your post in the command center because your shipmate is sad.
Uhura didn't abandon her post; she walked to the elevator to ask an officer where he was going. That's the kind of information she will be asked.
And in the subsequent case, she didn't abandon her post; they're allowed to leave the bridge when necessary, in the hands of other officers. And Spock, as commanding officer, seemed to appreciate it.
You know that's odd. When I served in the military you didn't abandon your post without proper authorization prior to leaving it. In fact it was a general order that you did not leave your post w/o permission.
If she was temporarily in charge, she could have left the post, but that would require she appoint someone, no matter how temporary, to assume command.