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If my rogue attacks while hidden and dual-wielding shortswords, do I have advantage on both of my attacks (from Two-Weapon Fighting) or only on the first attack?

This is similar to When attacking while invisible, when exactly does invisibility break?, and to this Sage Advice. There's also an informative Sage Advice podcast segment about stealth and attacks.

The answer is no; you reveal yourself on your first attack, unless you have a special effect that says otherwise.

Once you have made an attack, hit or miss, the character is no longer hidden. The only exception is characters with the Skulker Feat, which both greatly increases the places a character can hide (Light Obscurement is enough) and allows the character to remain hidden after missing with a ranged attack.

That said...

In combat, most creatures stay alert for signs of danger all around, so if you come out of hiding and approach a creature, it usually sees you.

Without some extenuating circumstances, you cease to be hidden before you make a melee attack. If you're able to make the attack, then you didn't have cover from the target enough to Hide in the first place.

For a rogue in melee combat, you're more likely to get Sneak Attack from an adjacent ally than being hidden. The ally doesn't grant advantage, but it is enough to allow Sneak Attack.

Other effects can grant the Rogue Advantage - darkness when the Rogue has darkvision, other forms of obscurement when the Rogue has blindsense (which they all get at L14), and so on. If the target doesn't have similar capabilities, the Rogue may get Sneak Attack. Being hidden doesn't matter in these case.

Either way, just in case it was unclear, a Rogue only gets Sneak Attack on one attack per turn, no matter how many they take. Of course, there are ways to attack on other people's turns, which could grant Sneak Attack more than once in a round.