Can you sue the person your spouse has an affair with?

No.

Texas eliminated this type of lawsuit called "alienation of affection" many years ago.

Instead be mad at your spouse. 

I always find it interesting that it's always the wife that is mad.  Your husband had the affair.  No hussy forced him to have the affair.  He knew what he was doing.  He was an ACTIVE PARTICIPANT. 

There is something wrong in your marriage if he had an affair.  You need to focus in on that and not focus on the other woman.  You need to be mad at your husband.  He violated your trust.  He violated your marriage vows.

Most of the wives defend their husband's behavior and get angry at me for suggesting that they participated in the affair!