Do I need to render a salute to an officer when my hands are occupied?

It is completely proper based on customs and courtesies, to give only a verbal greeting when both hands are occupied, especially in the performance of your duty. Obviously, discretion should be utilized, say if approached by a General or commander at the Battalion or Wing level command or higher.

BS.... SGT

Do your best to keep a hand free and not get into this situation. Aside from that yes no UCMJ recourse but is this why you want you chain of command to remember you?

If both your hands are full, you greet the officer with "By your leave, sir". That being said, you really should have ditched the coffee mug, and saluted the man.

That's life.
Put one on the floor methodically ,free your saluting hand ,perform your duty and drive on.

I would have done what I was told. DROP the coffee and binder right there, render a salute, shout out SIR, YES SIR! to let them know deep in their hearts what assholes you believe them to be. When I was onboard the USS Oklahoma City, CLG-5, I was taking a dump around midnight or so, in a stall on the back row of stalls. No doors on these stalls, mind you. So.. I'm trying to drop a mud dog and I hear footsteps coming near. I see movement, and there's an ensign looking in at me. I said loudly, "EXCUSE ME FOR NOT COMING TO ATTENTION, SIR!" He turned to walk away and said "Carry on.." I said loudly, "SIR, YES SIR!" and heard a little laugh as he walked out..

Am I missing something? I thought you weren't suppose to walk around with a coffee cup outside anyway, it's like walking with a cigarette. At least that's what I was told. And as far as inside, not sure? But leaving the cup and going back later for it. Or finishing it first might be a good idea. Either way leaving right hand free. Now as far as the dress down, uncalled for unless brownie points were in order? Just my thought.

Absolutely not. If both your hands are occupied, acknowledge the officer and give the greeting of the day.

If they are not cumbersome items like a binder and a mug then you should adjust them both to your left arm or put them down and salute. Making no effort at all is clear disrespect.

It sounds like you could have put the binder under you left arm and taken the mug in your left hand leaving your right arm and hand free to render the salute. My understanding is if your arms and hands are full that you must then pause walking and stand aside at as much in the attention stance as possible. maybe even offer and apology to the ranking officer of your inability to salute or if they are engaged in conversation with the other person, remain quiet and wait for them to pass.
You probably messed up when you greeted them. you may have interrupted a conversation and brought attention to yourself.