Can Truesight (as granted by the true seeing spell) see through the darkness of the hunger of Hadar or darkness spells? For hunger of Hadar, would vision be obscured by "milky, otherworldly tentacles"?
Yes for the Darkness spell, no for Hunger of Hadar.
As is stated in the PHB, pg. 185, truesight can definitely allow vision in darkness, both normal and magical, much as the Devil's Sight for Warlocks. Given that the description of the Darkness spell notes that it creates a magical darkness, creatures with truesight can see in and through it.
Hunger of Hadar is another beast altogether. It effectively creates a void that warps the fabric of space, and blinds creatures within the area. If a character is within the spell, then truesight does nothing as it doesn't protect from the blinded condition, which this spell applies. As for any creature with truesight trying to look in or through the sphere, this also is impossible, as the spell doesn't mask reality so much as create a gate to another plane that isn't superimposed with the material plane, as the ethereal plane is. This is also why no light spell, regardless of the level, can illuminate the darkness created by this spell. Basically, the darkness present isn't created by a spell, but as a side effect of the appearance of that dimensional gate.
The Darkness spell, yes. Hunger of Hadar, no.
The description of truesight on PHB p. 185, says:
A creature with truesight can, out to a specific range, see in normal and magical darkness...
Darkness is specifically stated to create magical darkness so yes, a creature with truesight would be able to see in darkness.
Hunger of Hadar, however, says:
A 20-foot-radius sphere of blackness and bitter cold appears [...] The void creates a warp in the fabric of space...
Hunger of Hadar doesn't create darkness (in the sense of game mechanics), magical or otherwise; it just creates a black void of space, so a creature with truesight wouldn't be able to see within that space.
As further backup, Jeremy Crawford - lead rules designer - has stated that an ability that's almost identically worded to the above quoted text for truesight isn't meant to see into the void of Hunger of Hadar:
Devil's Sight is meant to pierce the dark created by a spell like darkness, not the void of hunger of Hadar.
How can you be have true sight and be blind? What is darkness but the absence of light?
If i were the DM and someone had true sight then i would say they could still see in the void. Although in this case there is nothing to see but the otherworldly tentacles rubbing against you and whoever else might be in there with you.
They would also be able to see where the voided ending to help guide them out.