Hmmmm... Sounds like one of the typical beginnings of 3-11 shift on the Med/Surg floor where I work. I feel your pain and will first say that yes, the ICU patient transfer is your priority. Usually, I prevail upon the nurse from the previous shift who had the patient needing transfer to stay to help with the transfer and give report to ICU for the sake of continuity, and they've usually been fine with that-- It only makes sense.
I would then approach the upset/crying patient, introduce myself and quickly let her know I will address her concerns and problems as soon as these other urgent needs are attended to. I BRIEFLY explain the need for patients to go for their tests, to ICU, etc. I find that taking a short time to show a needy patient that I hear them and will eventually help them is worth the few minutes, and they usually calm down.
With patients leaving the floor for tests or procedures, I just do the best I can. If the report I just received indicates that they are stable, I will introduce myself as they are being placed on the stretcher or wheelchair and maybe quickly assess their lungs if they're in with a respiratory problem or check their incision if they're post-op, for example, and send them on their way. There's not a whole heck of a lot you can do when everything happens at once like this. I often feel like a pinball-- pinging and bouncing from room to room to telephone to family member to-- well, you obviously know what it's like, troublennc!
You also said: "What if some thing happens and I'm asked why this or that was not reported before the patient left the floor? I can't say well, I was too busy." Well, sometimes things happen and they are often beyond your control. The truth is, we ARE too busy, but it's like that guy with the plate-spinning act on the old Ed Sullivan show, trying to keep all the spinning china plates on top of the tall, wobbly sticks. (I'm sure this is WAY before your time...
) Once in a great while, one of the plates came crashing down, but you just have to do the best you can to keep the other plates in the air. I tell myself I am one person with two hands and two feet and I can only do one thing at a time. When I get nuts, I can't think *OR* prioritize, so I take a deep breath and ask myself "what is the next thing?" and do that.
Also, don't be afraid to ask for help and to delegate to your PCT's (aides). Coworkers respond best to direct requests (with please and thank you, of course) better than whining or responding when you're in melt-down mode. Believe me, I had to learn this the hard way.
I wish you all the best... It WILL get easier. Please be patient and kind to yourself as you learn and grow. We all were there once and I continue to struggle to deal appropriately with the kind of chaos you described.