Not sure why you're being downvoted. I'm pretty particular about grammatical correctness but I've come around to thinking that singular "their" is a reasonable approach. I wouldn't necessarily call "his" sexist but, especially when used in the context of certain occupations for example, it does perpetuate a stereotype. And interspersing random "hers" calls attention to itself and is distracting.

There's also precedent with thou and you although that evolution was a bit more complicated and isn't quite the same thing.