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

If you are able to move the mug to the other hand you should. But if your hands are full and unable to make that adjustment, then it’s not required.

Keep your rand free to render a salute, unless you are on a work detail or caring a single item that requires both hand to be sued.
SFC(ret.) R. Jay Book

In the Navy, such an Article only applies to the carrying of objects that make rendering a Salute a major impracticality, like carrying an object with both hands. Neither a mug, nor a cup in hand is consider impractical, as you can shift from your right hand or put down, those objects with ease. If you right arm or hand is incapacitated, you Salute with your left. Always render the proper verbal Honors/Respect.

Well, maybe someday the CSM will fulfill his need for power. Let it be a lesson that in any organization there is and will always be a need for someone to feel tyrannical. As your life continues you will remember this and it will make you a better leader.

Saluting is an honer between men (persons) of arms. If you can't or won't salute first, Sergeant: I will salute and great you. However this sounds like this is a habitual occurrence. Perhaps YOU should not carry anything in your right hand while walking about.

It appears that if you hold your mug in left hand you can possibly shift you binder to under your left arm, render the salute then grab binder into your right hand again. If it takes both hands to carry an object like a box etc then the regulations apply. In other words, if you're able to carry your items in a way that you can accommodate a salute, then do that. You're citing the regulation to us about when it is ok not to salute. When you're confronted with the lack of salute again, cite the regulation to that officer. Regulations take precedent over officers ego.

Your hand were not issued anything to hold. They should be free….

According to the Army manual, you do not have to render more than a verbal greeting when both hands are occupied in a laborious task and attempting to render a salute could cause danger or harm. USMC SOP was nothing in the right hand in garrison when saluting was appropriate. Drop blouse and cover when saluting was going to be dangerous.

I've been out a long time but I recall being instructed to not carry anything in your right hand. Otherwise on working parties, one Marine, usually an NCO would call everyone to attention. Everyone stops what they are doing, comes to as close to the the position of attention without dropping anything, and the NCO salutes.

Well, there is a solution: come to attention, call out, "by your leave sir", and await his physical or verbal salute. If he does not return a salute, remain at attention and call out, "by your leave sir", again. If that CSM is there, again, give the officer a verbal blessing, "Sir, have a blessed day". If the CSM questions your only giving to the officer, remind him you are to address the senior in the group, not the junior, and move out.