Okay then let me ask you this.
You’re the person responsible for everyone’s safety.
Evil Spirits come to attack and you and all of your friends and mentors have proven absolutely ineffective at dealing with them.
A guy comes along with a new technique (to you) that’s capable of dispelling said spirits, and it’s learnable by Waterbenders.
The Spirit threat looms.
However said guy seems shady and has extreme issues with your father for reasons not completely clear.
But he also would be willing to help teach you on your Spirituality issues (which are also diluting your strength in the Avatar State) and help deal with the threat. He also expects nothing back (or at least isn’t vocal about such a desire).
Your other teachers are adamant you do not agree with a proposition, and one would like you to leave the situation and learn a completely different skill irrelevant to said problem at hand.
Both sides have their pros and cons. Finishing learning all the elements is of course paramount to being a Fully Realized Avatar. On top of that the Dark Spirit issue has apparently been on going since sometime after the Hundred Year War and the Southern Water Tribe doesn’t seem too negatively affected. On top of that Unaloq is hated by pretty much everyone you know that knows him.
On the other hand, a low spirituality is detrimental to an Avatar’s ability. At that current level the Avatar State is at most a slight buff, not the world beater that Aang and most other Avatars are when they enter said state. Training with the current teacher is NOT helping in the least. It took moments completely separate from training to even find a sense of morality, Tenzin didn’t really help much at all. The Dark Spirit situation has gotten people killed, which the Avatar is essentially responsible for. On top of that, it’s not helping the North’s slight (if not more) bigotry towards the South.
Personally, I feel that as Avatar, you need to help people survive as much as possible as long as it wouldn’t put as many people at risk at the same time. Compare this to when Aang was asked to abuse the Avatar State to go after the Fire Nation immediately after Book 1. It would seem like a similar situation (though instead of a shady leader, it’s more of a stupid, douchey one) however Aang doing so would put more people at risk than just learning the elements due to at that time, the unpredictable nature of the Avatar State. I would try to solve the Spirit Situation and hopefully get some help in the Spirituality department. If Unaloq turns dick, he gets kicked to the curb and beat down with the quickness.
Another thing to note is…what could Unaloq do to really screw everyone? He already has dominion over the Water Tribes. And if he did something to screw Korra everyone would take her side because…she’s the freaking Avatar. And if anything happens to her the fury of 4 nations would come down on his ass. This isn’t like the issue from Book Air, where she was an unknown Avatar with a shaky reputation and in a political situation she doesn’t win by default (no thanks to Tarrlok ruining Bender perception). She has everything thing she could ever need right now to take down Unaloq if it came down to it. So it’s better for her to take the advantage of learning how to stop Dark Spirits while also hopefully getting more spiritually tuned. Going with Tenzin would be a crapshoot right now. Especially since she’s already shown that she’s an EXTREMELY quick study with Bending Forms so more Airbending help isn’t a pressing need at the moment.
This is all opinion based though until Unaloq shows his cards.