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

Normally I would say who cares. An attempt at proper costumes was made so good enough.

I think however it really depends on the situation. Would it have been easy to consolidate both items in one hand? If yes then you probably should have.

In this case it's hard to say. How big was the binder, how big was the mug, did it have a lid, was it full of coffee or empty, etc.

That's why I don't bother trying to "correct" people on situations like this

that is right

First of all, saluting is not a chore. It's a privilege. It's you demanding a return salute recognizing you both as professionals.

CSM just being a CSM. If your hands are full and you can't adjust then no need to salute, but do give a greeting to acknowledge the officer.

For one the mug should have been left behind. But this could go two ways. Should have adjusted to the left hand / arm. The officer could have replied "Carry On" and you great the officer with "Good Day, Good Morning, Good After Noon or Good Evening"

Nope! I reprimanded one of my lieutenants for chewing out one of my soldiers when the private didn't put down a box he was carrying and saluting the lieutenant.

A binder and a mug? Transfer the mug to your left hand and salute.

I was taught that your right hand should be free at all times

Most respectable Officers will acknowledge your situation and tell you at ease before you start fumbling

can you put the mug in the other hand? There is really someone that has time to worry about stuff like this? If I were you, I'd apologize and point out that your carrying these items. Bring your rule book with you figuratively at least and be ready to justify it as politlely as possible