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

I am reminded of a incident that happened at McChord AFB. A Airman was coming out of the PX with his 9 month pregnant wife. He had two bags of items along with a box under each arm. The commanding 2 star was walking towards them. She knew her husband couldn't salute so she walked up to the 2 star and snapped off a salute like she had just graduated from OCS. The 2 star returned her salute and without even breaking stride told her to," Suck in her gut!".

In my humble opinion, you should do your best to make your right arm unoccupied for the purpose of a solute. If you're holding a box however or something that is unable to Allow you to render a salute you at least have to acknowledge the officer with a good morning sir or ma'am. Or something to that effect. I haven't seen anywhere in the Regulations that talk about this .

depends what kind of an ass the officer is?

It depends on what you are carrying that occupies both hands. If you are carrying a bulky container or something that definitely requires both hands to transport, the answer would be no. In the circumstance that you describe, however, you should have transferred your coffee mug to your left hand while tucking the binder under your arm and thereby, you would have had your right hand free to render a salute. Had you been one of my subordinates, you would have been admonished to use better judgement from that moment on.

The right thing to do is just acknowledge people. I’ve never in my career been focused on rank when it comes to these situations. The most important thing is to always do what’s humanly possible and during this situation is give the proper greeting of the day no matter what rank they are.
Even when not in uniform I always acknowledge people and you wouldn’t believe what kind of doors it can open for you. Common curtesy is an uncommon virtue in today’s world.

No, as long as both hands are occupied (and the right hand cannot be freed up) render the proper greeting of the day.

Follow the regs ,,

I believe it is military protocol to render a salute to the officer. Take necessary steps to free you right hand and perform the salute. We as soldiers all had to learn . This is not a difficult task.
1Lt William J Pieczara (USAR ret)

When in doubt always refer to the regs. But as a subordinate soldier you should attempt to always be respectful of the rank, ie... move coffee mug to left hand and binder under the left arm and render the salute. You not only show respect but the strength of your character and it will come back to you favorably down the road.

Aren't the regs clear on this? You acknowledge the officer and carry on with your day. The officer should know the regs as well.
As as far as not carrying anything in your right hand. Well, most people are right handed so maybe we should salute with the left instead.