how do I logically counter the "philosophical zombie" problem?

1. Occam's razor. How is it that only you would have "real" consciousness/self-awareness while everyone else was just a really good automaton? Such a situation would require a much more complex explanation, with many unproven assumptions, than if everyone else were conscious like you. And who made those automatons? For what great conspiratorial purpose? Why go to all that trouble?

2. MRI scans. You could take scans of other peoples' brains, and of your own, and verify that they function just the same.

3. The creativity of others. How is it that a philosophical zombie can write meaningful prose, discover great new scientific discoveries, design cars and computers, create great works of art, or reflect on deep philosophical issues? All these endeavors require stupendous amounts of creativity, far beyond what any computer program is remotely capable of. (And then, who wrote the programming for all the philosophical zombies?)

4. Who created you? Are your parents philosophical zombies too? If so, how is it they were able to give birth to an actual conscious, self-aware entity?

You could play with solipsism, but that's usually just playing rather than really trying to believe it. I can't see how you prove self-awareness. When people start trying to compare human thought to animal thought, it gets really hard to draw the line. I have heard that "insight" is the difference between human thought and animal thought, but occasionally you see evidence of new "insight" in animals as well.

I think that if you want to break it down logically, you need to start by defining the terms. I have a feeling that you will find that you have no objective way to define the terms, and that the way you define them will aid your "proof". Detractors from your idea would attack your definitions to make the proof seem meaningless.

If other people are not consciously aware, then it's a matter of perspective. You were raised by these creatures, fed, taught, given medicine and technogy, and you've interacted with them as if they had sentience for your whole life.

If it is all a ruse, then the deception runs deeper than your capacity to decipher, and evidence to the contrary would be subject to the same doubt that might cause you to question the authenticity of it in the first place.

So much of your ability to rationalize is built on the concept of 'others are sentient', that assuming otherwise would be detrimental to your ability to navigate your environment. Other's sentience is fundamentally built into our language. The assumption is unavoidable.

Blow their heads off with a philosophical shotgun.

How do you know that you are not the zombie and they are just smarter than you and don't want you to know it?

Two easy answers:

1. cognition is impossible without consciousness.

2. knowledge is impossible without consciousness.

In fact, raising the question you just posed presupposes your consciousness.

Consciousness and self-awareness are axiomatic givens for homo sapiens. Only some folks in academia would assume otherwise, often as part of an intellectual game that has little puprose or goal aside from the game itself.