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

Others have nailed it but truth is in some circumstances you’re correct, and in some the CSM is right. Hard for us to choose a side in your specific case without having been there to see it. A mug and a binder? You probably could have had reasonably on your left side to keep the right hand free, and should think in such a way. BUT in cases that both hands are truly, reasonably occupied, a greeting is absolutely appropriate without a salute. Reg clearly supports such. This, btw, coming from an officer that was stopped and “corrected” by a SGM years ago at Bagram because I merely greeted the passing Colonel with my hands legitimately full. I was pretty well told I was wrong for carrying so much and being unable to salute.

I agree with the gentleman who said that one needs to keep the right hand handy to render appropriate salutes as appropriate.

The day I can't hold a mug and transfer a binder to my left side, is the day l will renew my membership to the base gym. I'm torn between calling the soldier ignorant or defiant.

WHAT BUGGED ME WAS SALUTING A CAR WITH AN OFFICER STICKER ON THE WINDSHIELD WHILE A KID WAS DRIVING THE CAR I DONT KNOW IF THEY STILL DO THAT I WAS IN THE ARMY A LONG TIME AGO.

TECHNICALLY that's why they don't want you carrying things in your right hand if you can help it. Ive dealt with this both ways. Junior officers get pissy from time to time about the proper respect thing. Older officers understand when you're just trying to get on with life. My CO used to pre-emptively tell us not to get up or stop what we were doing just because he was walking by. He'd say "time enough to salute me later."

No you don't. You recognize the officer by saying, "By your leave sir." I have however seen a few butter bars, REMFOs, and the so called 3rd Lt's. go ballistic over this. Most officers are not idiots.

In 1988, I had my daughter in my right hand and diaper bag in the other at 97th General hospital.. A Major was coming the other way, He saw that I was trying to juggle my load to salute him.
I recognized him and acknowledged him. He said that was OK that I could not salute.

"Do I need to render a salute to an officer when my hands are occupied?"
In a legitimate situation, no.
However, you mentioned carrying a mug … were you walking outside, in uniform, and drinking coffee? I thought that in and of itself was against regs. (Drinking, smoking, etc) Those who know better please speak up.
And to define "legitimate situation": in my mind, it's carrying items necessary to accomplish some legally-sanctioned military activity.

I passed a General, 4 star, in Vietnam and said good mornig Sir. He replied good morning soldier with a friendly smile and we both went about our business.

I was at my first training command right out of boot camp back when dinosaurs roamed the land. The Army post had all services there for training. As a Seaman Apprentice (E-2), I was on casual duty waiting for my class to start. One day I was carrying a box of the long, fluorescent tubes from one building to another and passed an Army Captain. Since I was carrying the box with my right arm, I saluted with my left hand. The Captain returned my salute, but a nearby Army SFC saw what happened and steamed up to ream me out. Just as he got to me, he stopped himself when he remembered that Sailors can salute with their left hands. He started muttering profanities and stormed off.