Yeah this happened to me last night. Finally it stopped after I started shooting the Klingon ships.
Having the same problems as the OP as well. Pretty much exactly as he is describing; May have something to do with the lag/strain on the server/cluster.
Make it as the chatbox is, so that you can minimize the dialog and reopen it later on. Also the help can be minimized and is seen as yellow starfleet insignia on the right side of the screen.
What's happening here is that all of the team members' dialogue boxes are interacting. So if you've closed it (but your team member is still reading), the box will pop back open to the answer to whatever question they asked.
So just sit back and wait for them to finish reading everything.