Merriam-Webster has an entry for half-orphan, meaning someone with only one living parent. They say specifically a child, which would match the usual usage of "orphan" (as mentioned in the question), but it might be used of an adult sometimes, either jocularly or by extension. This doesn't distinguish which parent is dead, but could be combined with a further explanation if it matters.
"adult children" is sometimes used in contexts where age is important, such as a form requiring someone to list all children under 18 and all adult children living with them. And someone might use it to emphasise that their children have left home or aren't dependent on them. But you wouldn't introduce someone as "my adult child/ren".