Looking at the timeline of Flynn's actions, it seems likely that Flynn had been in negotiations for immunity for at least the last two weeks. The fact that he went public with his request would indicate that he didn't get the answer he wanted. I do think his lawyer coordinated the public release of his request as a last ditch effort to force the Justice Department to give him immunity.
Right. This in my mind throws water over the thought that Flynn is trying to "undermine" the investigation by getting immunity and then revealing nothing. He wants immunity because he has legit fear that his testimony will have consequence in some manner.