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

During 1962-65 I served aboard a submarine homeported at the DesSub Piers at Navsta Norva.
Some of the very junior enlisted tincan sailors made a game of avoiding saluting submarine officers by suddenly becoming very busy with tucking in the tails of their dungaree shirts with their right hands when they saw the approach of a submarine officer on a sidewalk.
Usually I ignored that ploy and carried on with my own life.
One time, though, I stopped the hotshot seaman on the waterfront and told him that I expected a salute.
He replied, "You'll need to wait a minute,"
and I responded, "I'll wait."
He finished tucking in his shirt and then saluted.
So I returned his salute and said, "Thank you; carry on."
Then he carried on, and so did I.

You are walking past the two with both hands occupied. Always try to keep your right hand vacant but in this case that wasn't practical. A verbal acknowledgement as you pass is appropriate.

Bear in mind this is the Marine in me answering. It would depend on what I am carrying. If I am a Pall Bearer with a Casket, they can bite me. A binder and a mug are easy to set down carefully and render a Salute. All of that in mind, I may also consider if I know the Gentlemen. If not, give the consideration. If I do know them, acknowledge them and see if they offer to skip it as your hands are occupied. Do not assume it is o.k. and a CSM will always want the respect owed to be paid.

Trigger warnibg: I was active duty USMC in the 80s.

When not participating in drill, the right hand was to be kept available for saluting.
Beverages were not to be transported from building to building, especially open containers.
If an item to be transported would take both hands, a second SNM (preferably higher rank) would accompany the bearer in order to render any necessary honors.
Officers were told not to approach working parties unless they wished to participate.
Senior SNCOs, while accompanying officers, kept in mind that while the officers were in charge, without the enlisted nothing would get done. Attitudes were kept in check and officers were trained accordingly.

Render the customary verbiage and move on.

I retired in 1983, so I’m looking at this from way back when. If you have something in your right hand, you cannot salute. Why do you have something in your right hand? Would you, should you, what a responsible person in your situation remove the thing from your right hand so you could render a proper salute.
In my opinion. If your right hand is busy trying to prevent somebody from bleeding to death, don’t salute. If your right hand is holding a cup of coffee, get rid of the coffee and salute properly.
The salute the basic foundations of the military. Like any secret handshake in any club, it’s very important, but not the beginning and end all of everything. Common sense on both sides should prevail. Not knowing all the details. It looks to me like you went out of the way to try and be unable to salute. That was disrespectful to yourself, your unit, and the military in general.

If you are on duty and both hands are full, especially where safety or the pursuit of your duty is involved, you are not required to salute (and officers are not required to return a salute).

But I'm also not giving legal advice, if you need legal advice, consult an attorney.

Coffee cup? No wonder the CSM chewed your ass out. You're so in love with your coffee cup that you couldn't put the binder under your arm and the coffee cup in the same hand? SMH.......

Don't carry items in your right hand and always salute in garrison.

You had a binder in one hand and coffee cup the other and you didn’t think shift your load to free your hand for salute. The CSM was, in my opinion, correct