This is amazing. I'm a classical guitarist and pianist (not any good, but it's very clearly the way I'm oriented). Jazz just seems to escape me, although I work much harder at it.
One of the things that has really helped me with jazz is going through the older standards and singing the voice line while comping the chords on the guitar. Not only is it super fun to play and sing at the same time, but it gives a much better understanding of why the chords are there and what they're trying to do.
I have tried to reverse the two (play classical guitar, and improve piano) and I simply cannot, or I look like an absolute beginner.