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

The Correct Answer, is you should never have anything in your right hand that prevents you from rendering a proper Salute, with the proper greeting of the day.
AS a E-5 in the united states Army you should all ready know this.

If you have decided to carry one item in each hand , thinking you are being cleaver by finding a loophole that gives you a valid reason not to render a hand Salute, you are mistaken, you should always be prepared to take whatever is in your right hand and move it to your left hand, in order to give a proper Salute, with the proper greeting of the day, it not that difficult to do , it takes just for a few moments, then you can move the Binder or mug back to your right hand.

on the other hand, if you are carrying something to big and Bulky to easily move from one hand to anther, because both hands are full, then you are allowed to just give the Greeting of the day.

if the CSM Insist that you give a hand Salute, yet you still feel very strongly about the issue, I Recommend you give the proper Salute, with the greeting of the day, before you start asking questions for your Clarification on army regs .
After you have rendered your hand Salute with the proper Greeting of the day, you can Calmly, Politely, with all military Customs and courtesies and professionalism, explain to the CSM , That you meant no Disrespect to him, the CPT, Or the Army, that you truly believed that you were following Army regulations as per AR600-25, and seeing how you are working on making Staff SGT, you would greatly Appreciate a education on why you were Mistaken.

If the CSM gives you a satisfactory answer, then all is well, if the CSM gives you a answer that you disagree with, I recommend being very carful on how far you want to push, some battles are not worth the headache they bring at a later date.

like I use to tell my son, when he was learning to drive, and he would complained about anther driver cutting him off, or the driver would run a red light, My Son would say he had the right of way, according to the drivers hand book he was in the right.

I would explain to my son, yes he was right, but did he want to be seriously injured or dead right? as per the driver hand book? it is best not to have road rage.
there a time and place to pick these battles. they are not to be waged in public,

sometimes it best to let things go, wait for the right time to speak to the CSM , about the issue

you are most likely asking yourself, what does this story have to do with when to give or not give a hand salute?
the answer is simple , is this a issue , a battle or hill worth fighting for?
will it be worth being on the CSM Radar?
once you are on the Radar of senior leadership, they start looking very hard at everything you do, they start looking at your every action, on customs, courtesies, & leadership, etc this can effect future advancements & promotions .

do not misunderstand me, I am not saying not to ask legitimate questions, this is how you learn to become a better service member, but if you are just trying to find a way of not having to give the proper salute, then don't waste your time.
I myself have delt with many Junior personal playing fast and loose with regulations thinking they were being smart, it only bit them in the backside.
just remember you might win this battle , but you most likely will lose the war.

Pretty sure you just give the greeting. Or stand in front of the capt and CSM make a big show out of trying to hold the coffee and binder in the same hand while spilling the coffee over the CSM.

I feel like if you had made an effort to put the mug on the binder and salute, you would have been better off. I used to smoke and carry a mug and always found a way to free my right hand to render honors.

In today,s military some try to change traditions that are needed to maintain the discipline the military must have to complete their mission.

A lot of BS about keeping your right hand free... If you are carrying multiple items, carry them safely - in both hands! AR 600-25 prevails! If you find yourself in a position where you need to render a salute, if possible, quickly shift items to one hand (balance the coffee cup on the binder) and render the salute. If this is not possible to do, nod your head and state "Good Morning Captain!". The CSM probably thought you could have safely freed up the right hand and saluted. It is difficult to get to the top NCO ranks if you are constantly making petty calls to Jr NCOs.

Second time... all of my training and emphasis on customs and courtesies allowed me to want to follow the spirit of the regulations and courtesies. When I was going to an outside area with a great probability that I will cross paths with an enlisted or senior officer, I made every effort, while inside, to be prepared to render the proper greeting, especially to to those with lesser rank who were going to initiate a salute. My intent, my desire, my spirit was to follow the rules. SGT Martin, if your general intent and desire is not to be respectful and not follow the spirit of the courtesy, then you might already know your answer. -- I have been retired for 19 years and have a full beard, but I still have a quick moment of regret when the Fort Sill gate guard salutes me. I know it is alright, because even active duty officers on leave might grow a full beard and are still saluted. It is just the spirit of the situation gives me a quick moment of uneasiness. -- The AR allows for not saluting when you are genuinely on task using both hands for a bulky items. My pet peeve on active duty was not putting your hands in your pockets, except to place or retrieve items. I did not brow beat folks, and my soldiers knew it was a thing for me, so they generally tried to not keep their hands in their pockets. In fact, a few of the worst offenders were standing around in the motor pool just relaxing and talking; I happened to be walking up the stairwell that had a window looking down to where they were and without knowing I was around, I noticed one SSG begin to put his hands in his pockets, but immediately pulled them out. I knew my work was done at that moment (smile). The second occurrence was after a training exercise, most everyone had finished cleaning their equipment and vehicles. I noticed the commanders driver was not around, I went to the wash rack, he was the only one there washing his vehicle in the cold. He stopped for a moment and put his hands in his pockets to warm them. He did not see me, but as I approached, he turned around and quickly took his hands out of his pockets. I quickened my step to get to him and said, "Sometimes common sense has to prevail, it is ok to warm your hands in your pockets." -- SGT Martin, hopefully the officers in your community are indeed worthy of your desire and intent to follow not only the rule, but the spirit of the rule. More importantly, I hope you are worthy of your officers desire and intent to allow you the grace when you are not able to follow the rule and spirit of the rule in any military situation.

I would remind you to read the UCMF. Before taking advice from anyone.
It is my opinion that you do not have to salute an officer if both hands are full. Make eye contact and nod your head.

No you don't have to.
You are 100% right.
PO3 William E. Carroll

I would just say, I would salute you sir, however my hands are full sir.

No, you are not required to render a salute whilst your hands are full.