I have 20 yo daughter from previous marriage and my current husband want to adopt her to give me and her his medical insurance. She is fulltime student in college. How we can do it? Where do we have to go?

Because she is an adult the 3 of you ( really only the 2 of them ) need to go before the courts with adoption papers and make clear that she consents to this. There will be no more rights or legal ties to the natural father. She needs to be completely aware of that. No inheritance etc. You can find a local lawyer in your area that should be able to handle it pretty cheap as it is a slam dunk with willing parties involved.

If you are married, she and you would be eligible for his insurance anyway.