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

If it was one of those things that caught you off guard then I can see the point but honestly no matter what my rule of thumb is that if I can spot an officer rank from about 15-20 feet “being aware of your surroundings” and am carrying things in my hands I’ll try to move all items to left hand to issue a salute. If your hands are full of like heavy objects like ammo cans or like weapon parts just issue treating of the day or I think it also depends on who they are and if you know them.. ie PL, CO, ect or it’s some random full bird walking around with BDE CSM. If that was the case I’d prob just stop, set my things down go to attention and salute. Once they rendered it back and passed pick up my shit and move on. It’s about showing respect and not trying to be coy or lazy and having situational awareness.

In response to your original question.
Long answer: Probably
Short answer: Yes
Proper rendering of respect to rank is more important than your mug of whatever.

I was on crutches for my last 2 years in I still would stop and offer a salute most officers would offer a comment and tell me not to, but I still made the attempt, it sounds like you were looking for a way out and it was obvious and ghats why you got the well deserved reprimand.

As a butter bar in QM BOLC I saw another butter walking to class with his coffee in his right hand.

I mentioned to him he should probably switch hands.

No sooner did he blow me off we passed a USMC Captain and seeing the other butter was covered in gasoline the good Captain proceeded to light him up.

Similarly while still in BOLC I was walking out the building, and some old guy (coming from me that's saying a lot) was in a hurry to get in the building and both his arms were full (with his ASU still in the cleaners bags). Before I realized he was the School Commandant he had come and gone and nothing happened.

Regs or not, some folks seek out reasons to harp on things, others have actual things to do.

CSM's are a special breed. You should know that as a SGT by now.

A nod and appropriate greeting should suffice. good Morning Sir, Ma'am. Good morning SgtMaj. It's the CSMs job to preserve customs & military courtesy, but unless it's for Colors work takes priority. Texting does not count toward that, so it depends.

Your shouldn’t be walking while smoking or drinking

In my experience the on paper answer doesn't always satisfy people who are looking for errors or deficiencies. I've found it easier to just salute someone and while you're shuffling your belongings they'll tell you no need to salute.

The right answer would be don’t carry the mug of coffee. Or figure out how to carry that binder and your coffee in your left hand. You could have walked with the binder under your left arm and you’re mug in the left hand leaving free your right hand. Your an NCO don’t make excuses as to why you can’t adhere to proper customs and courtesies.

IMO you should keep your right hand free when in an environment where saluting is required. Your description of the situation sounds weak. Could you have tucked the binder under your left arm and held the mug in your left hand? Maybe the mug should have stayed at your desk or kitchen? You’re an NCO. Be a good example!

Quit playing “hands full” games and render the salute. From your question, it looks like you have a habit of ensuring your hands are full to avoid saluting. Looks like I just found the instructor for next month’s NCOPD class on military customs and courtesy.