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

Do not be an idgit. Carry binder and mug in left hand. You are to salute with right hand at any time needed. Somebody needs to take away this soldiers rank and send him back through boot camp

This has been addressed before. YOU are looking for an excuse to not render a long held tradition of saluting officers. You need to recognize your place, the oath and commitment that you agreed to. The alternative is to become an officer and either be an a-hole to enlisteds like you are or be forgiving when they are deliberately carrying items like you were just to be an a-hole against officers.

Your error, SGT Martin. You're required to leave your right hand free when you're in saluting territory.
(Or, see LTC Burch's comment.)

If you are holding something in BOTH hands and don't salute me and acknowledge me, then I will return your actions. If you have a cell phone in your right hand and don't salute, I may well discuss the issue with you - unless the Chief does it for me.

Yeah brother a mug and binder ain’t gonna cut it. The only way I ever got away with not saluting an officer was when I was moving heavy boxes out of the unit and even then I still acknowledged them. You could’ve put the binder under your left arm and switched the mug to your left and rendered the salute.

I don't know the full circumstances, but a couple of thoughts come to mind. The first is that the CSM might have just been being a dick. But the second thought is that since this is your second incident, you might be seeking conflict. My advice is to consider your actions in light of what they might be implying. Not saying you were in the wrong, but any time you find yourself in conflict you need to examine your possible culpability - especially if history repeats itself.

If you can't hold a binder and a mug in your left hand and crook of elbow or under shoulder, you don't belong in my military.

I retired as an 0-4 but had 16 years enlisted before commission. I was taught not to carry anything in your right hand. I practiced that advice through my entire career up until I aged out at 62. As an officer I always knew there was a higher ranking officer just about anywhere you go so I kept my hand free to salute them as required. Equally or more important was returning salutes. I saluted everyone as if they were wearing four stars. It is an honor to salute no matter who salutes first. My feeling and my experience is that the people who have problems saluting or with the respect given both ways usually don’t excel and they are usually the person who does all the complaining and very little of the work. I found that to be true of both officers and enlisted and I have experience on both sides.

If you were carrying something that needed both hands, you would have been ok. However, it is easy to carry a mug and binder(s) on the left side with the right hand free.

Oh, lord no! Full hands are understandable, especially with babies or toddlers! A nod of recognition is sufficient. And in one situation decades ago, "I don't blame you, corporal. Were I you, I'd be holding hands with her myself!"