Short Question:

Shapeshifters are dual. If you have access to a natural weapon, you can use it at astral combat and depend on your physical attributes instead on your mental ones. But can you simply attack with the close-combat skill instead of astral combat? What if you are in your human form? Your aura is still an animal (which has a natural weapon). Does this qualify for such an attack or would you need to shift?

Depends on opponent.

p. 315 "Astrally perceiving and dual-natured characters
use their physical attributes and skills to fight opponents
with a physical body, and their Astral Combat
+ Willpower to fight wholly astral entities."

and

p. 399 "A dual-natured critter with a melee Natural Weapon
can use this power against astral targets that are within
its reach. Use the critter’s normal Unarmed Combat skill
and physical Damage Value for this attack. Ranged Natural
Weapons, like other ranged combat, do not work on
the astral plane."

When resolving rules you tend to use a specific > General

so in this case a Shapeshifter with natural Weapon would use Unarmed + Agility versus wholly astral opponents because the Natural weapon says so.

however, I would also allow the shapeshifter to use Astral Combat + Willpower since they can do that without Natural Weapon.

Shapeshifter can use either attack.