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

Regardless of the opinions, the regulation is pretty clear on this. The greeting of the day is appropriate. I remember when I was a PFC I was trying to adjust my items into one hand to salute a CPT and he told me that I didn’t need to and to look up the regulation and report back to him what I found. I liked that he made me look it up myself rather than just take his word for it.

Depends on the situation, if you are working and can’t surrender a salute. Please give the greeting of the day. The reg is specific and it states that one doesn’t have to surrender a salute. But, common sense would dictate. But, I guarantee you some officers don’t care about the reg or even know the reg. They will have some issues with it. Just be prepared to salute if you need to. As NCO’s it’s our job to know and sometime we have to educate.

Too much time has passed for me to know what the proper courtesy is. All my salutes these days only require 1 finger.

Address them as sir or Ma’m if pronouns are safe to use and say good morning or whatever time of day it is. A good of officer should respond with the same. If they demand you drop everything just for a salute they will lose the respect that is due them. That’s my opinion

The answer is that you shift what is in you hand and salute. Sounds like you are making excuses for not rendering military respect for an officer. You're a Sargent and know better. Start acting like an NCO and stop making excuses. Your lack respect is either poor military training or lack of respect which makes me believe you should not be holding the rank you have.
If you were carrying boxes and on a work party that would be a different thing but a coffee cup and a folder just shows disrespect for the service, its discipline and your nation.

Salute, or not to salute. Luckily we didn't have that "problem" in Vietnam!

A binder and a mug?
Are you f@#$ing serious?!
Binder under your left arm, mug into left hand.
Your handS were not occupied, you were trying to get around the rule. And you were doing it in a way that is not intelligent, and being a 35F means you better be intelligent.
How were you not taught this in BCT? You probably were, but you were imagining yourself beating the system and daydreaming when they said it, huh? If you are easily capable of having your right hand being free, it is not occupied. If you would have to stop and put something down to salute, that's what it's for.
Do push-ups.

Looks to me like you’re deliberately carrying things in both hands to avoid rendering the most basic of military courtesies.

Best response I ever had was when I was carrying a number of boxes full of parts down the pier and an officer stopped me to ask if he could lend a hand.

No, you are not required to salute when you are under a load, carrying something. I was walking across base with another petty officer and I had in my hands a box, he and I both passed an LTJG who asked him, "Is something wrong with your shoulder, petty officer?" My buddy had been in surgery for a dislocated shoulder and he thought the young officer was from his former command and held out his hand to shake hands (yeah dumbass) the officer got pissy and said "You are suppossed to salute." At which time he did. I just stood there smiling as I understood regulations and did not have to put down my load to salute.