Welcome to MB.

The steps to recover from an affair are the same, whether the OP was a cousin or not. You need:

1. A complete end to the affair. Complete no contact.
2. A change to the conditions that made the affair possible.
3. Transparency going forward, so that you can see each other's communications with other people.
4. An interdependent marriage, where there is no space for a secret second life, and where you meet each other's most important emotional needs and avoid love busters.
5. Exposure of the affair to critical people.

(There is actually a check list that we use that is more detailed than that, but I don't have it to hand. Someone will post it soon.)

The main thing that I can see your husband has not done is to end all contact with his cousin. If he doesn't do this, you, his wife, will never recover. You could achieve this, and exposure at the same time, by telling close family members about the affair, and explaining that you won't be going to family functions if she will be there. Explain also that this is why you are planning to move.

And then actually move.