Check out
Gemlab on (surprise) 47th street. They purely do appraisals, no selling. If I remember correctly, they charge $75 per carat or somewhere thereabouts.
And yes, you can go through the TSA line just fine with it. Just open the box when your stuff comes out on the other end of the x-ray machine to verify its there, or unless you're carrying around the Hope Diamond, just leave the stone in your pocket as you walk through.
I'd also definitely make sure its insured before I travel with it.
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might there be other cities or places where I might sell for top dollar
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There may be small market fluctuations from city to city, but by and large the prices you'll be offered for your diamond are going to be fairly uniform. I'm sure you're aware that you'll get a fraction of whatever the jeweler is going to turn around and sell it for. Do you have a GIA certificate for it? If not, get one, or be prepared to be offered peanuts.