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

That is talking about needing two hands to carry an item - ie a large box. You would have been able to have tucked the binder under your arm - and hold the cup in your left hand. - On the other hand, were you carrying those items as you did, just so you would not have to salute?

If what you wrote is correct, the CSM is a jerk. My problem is a binder and a mug. What are you doing drinking and walking [especially in uniform]. I was taught not to eat or drink outside while walking--it can get messy. Walking down the hall or a "no salute" area is different story.

Whenever I see an officer approaching, I stop what I’m doing. Render the proper salute, then continue on after the officer has proceeded on his/ her way.

Greet them with the proper military respect and move out. That CSM was being an asshat.

Upon seeing the approaching officer he should of stopped and placed the item in his right hand on the ground and come to the position of attention and held the salute till the officer has returned the salute.

Some officers are full of themselves. I had a similar experience. I had books under both arms and got chewed out for not saluting one officer. The other officers I crossed path with never said a word.

Don't argue just salute and keep moving.

Its generally understood that if both hands are being used that you simply acknowledge the officer. Nevertheless, that is an understanding, not a policy. If the officer needs you to render a salute, then you put down whatever you're carrying or stop what you're doing and render a salute. You always follow orders even when its not expected, completely unnecessary, and makes no sense. Some officer and even more NCO will purposely put you in harms way for no good reason other than to satisfy their fragile egos. Just keep in mind the if you're hurt in the process, the service will pay for your recuperation, and you may receive benefits for any lasting ailment caused by doing so.

if you are doing meaningful work and your arms filled , than you can say by your leave. It should show your tribute to the officer, just don't make it a habit. a coffee cup may not be seen as meaningful.

You need an attitude adjustment. If you use this as an excuse to salute which is military courtesy, I am pretty sure it will become something else in the future. Put the stuff down, salute, then move out. It's what soldiers do.