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

"Carrying articles" means something requiring both hands to carry due to size or weight, not because it is slightly more convenient. Honestly sounds like you're trying to get out of rendering honors by skirting the rules. Do you run from colors too? Military traditions are important and are in keeping with good order and discipline. Don't be that guy.

Common courtesies would be to move both items to the other hand or put the binder under your arm with the mug in your left hand and render a proper salute to the Captain.

Back in the day, if at all possible, outside one kept nothing in the right hand, or if you had to carry something with both hands, verbally greeted the officer. Had I been in that situation SGT Gianna described, I would have laid things down, come to attention and saluted the CAPT O6, and left the CSM to the lack of mercy of the CAPT. Perhaps the Navy of yesterday viewed courtesy as primary and how that courtesy was shown was open to a situation. Of course, as a Chief Petty Officer, I wonder if even a Command Sergeant Major would have corrected me?! But things were different 50-60 years ago. (I will comment I worked mostly in cross-service Navy-Army-Marine situations and assignments mostly.

The way I was always told it was if you were carrying "official" stuff, like tools, files, books, your gear, then a greeting was sufficient. But if any of what you have in your right hand is "personal," you stack it on top of the other stuff or drop it and render your salute.
But the only unreasonable officers I ever met were brand-new Butter Bars. I always saluted them, no matter what, even if it meant setting something down on the wet ground.
I've been out for 17 years and I still very seldom carry anything in my right hand.

WOW! Some people need to chill and have forgotten the entire point to a salute. The salute is an acknowledgement of the rank and status held by that person. It is NOT a salute to the person so, stop being so contrary people. However, a coffee cup is a bit ridiculous to say thats "occupying" the hand. If you do continue that, make a play at juggling things around so you can salute. The majority of the time they'll appreciate the acknowledgement and wave you off with a smile. That's just one of the sucky parts to being enlisted. We have to coddle some with their entitlement.

Poor baby, are they picking on you?
Stop walking around with a mug of coffee in your mitt.

just wink at em that ought to get their attention-----

You're a Sergeant and feel you need to ask a question like that...

If a soldier had his hands filled and walked past me, I did not expect or demand a salute. I did expect a greeting, such as 'good morning sir'. I believe the CSM was in error.

Carry the AR with you at all times and show it to CSM when fvckes with u. See how that goes