I think they are both correct without "again" at the end. If you add again to the end it implies that you have already done it at least once.

The first sentence is correct. The second sentence would be better as: "I promised that I would never do it again." Intentions have two form：to-infinitive or that-clause The first sentence is the to-infinitive form The second should be the that-claus form

Both sentences are correct but feel incomplete... You can add "again" at the end. I promised never to do it again. I promised I would never do it again.