Is there one word or a better way to say or refer to one's "adult children"? I have never found quite the right word, and "offspring" doesn't really say it.
"That couple has two ________: one is a lawyer and the other is a politician."
The word remains "children" regardless of their age. You can elaborate on that with "grown-up children" or, more awkwardly, "adult children" but if you're explaining that somebody's children are a lawyer and a politician then their adulthood is implied.
Child: son or daughter of human parents.
The most common and best way to refer to your adult children is
That's it. The OED entry has 16 primary senses for child but there's two main ones
A newborn child, esp. a girl; a young person below the age of puberty, esp. a boy
The offspring of human parents; a son or daughter of any age
This isn't confusing at all. Given that everyone is the child of someone, He's a child means that he either is a boy or acts with the petulant selfishness of one. He's my child means he is the offspring, issue, get, fruit &c. of my loins in union with someone else's. Adding adult just makes it weird and offputting and, if anything, pushes it back into the other definition: you're making him sound childish instead of like your child.
If you just wanted to sound lawyerly, though, the legal jargon is usually issue or, where grandchildren & al. are included, descendant. You shouldn't talk to or about your own kids that way, though, unless you want them to think you've been talking to an attorney recently and they should be nervous about the contents of your will.