If I found out after my divorce that my husband had been having a 3 year affair, can I take him back to court for alimony?

As a general rule, no (I'll discuss exceptions below).  You would have been able to ask for discovery regarding other women or assets related to gifts to other people.  If you agreed to the entry of the final decree, then you potentially waived your right to explore these issues.
The other issue that you have is that Texas does not have a statute for alimony.  Even if you went back and proved the affair, the court could not automatically award you alimony because Texas does not have an alimony statute.  Texas does have a provision for marital support-- in very limited circumstances where a women is not able to support her basic living expenses.  However, these types of awards are completely up to the discretion of the judge-- and again-- are not based on the existance of affairs-- but rather the needs of the spouse.
If your settlement was somehow linked to a pre- or post-nump agreement that penalized one party for an affair, then you may have a chance of reopening the case becasue you would have a potential right and subsequent remedy.  If you think that the property division was not just and right, then take your decree with whatever discovery was done in your case, and see what loopholes, if any might apply more specifically to your case.