This should die down if they just got into contact recently. How long is "recently?" If this has been happening for a week that is different than if it has been happening for a year.
My guess is, after she visits this will die down - if only because she won't be as interested. This seems like kind of what your husband does - he attaches compulsively in an inappropriate way. But few 18 year olds want to spend most of their time texting a parent, although in her case it's probably assuaging some of the pain of his abandonment of her.
Truthfully, while I understand how this is aggravating, this is a HUGE deal. A father and daughter reunited? Oh my gosh. That's a wonderful thing to be encouraged. How do you catch up quickly on so many lost years of knowing one another? I would be VERY patient and if it is a deal breaker for you to have his adult daughter a very significant and frequent part of his life, I find that problematic. Now, I don't know the particulars of why they lost touch at 6 years old . . . you say a custody issue. But as a parent, I'd figure out a way to see my child . . . which he did not. Maybe the mon is tough and made it very difficult. Don't know. But what I DO know is that this young woman was RIPPED off from a very significant figure in a person's life.
I personally would love your partner through this and encourage him to be a man of strong character. Putting a child he lost touch with as his priority for a while is not too much to ask or too much for someone who loves that person to give. In my opinion. good luck
I agree with AnnieBrook, RockRose, and SpecialMom. I would suggest that He may be lying and hiding His texts with His Daughter because You are expressing such dismay - He might think it's easier to lie to You than to argue with You. I too, think it's a wonderful thing for Him to be re-united with His Daughter and it would be just as wonderful if She were to be welcomed by His Wife and 2 Sons.