Google hangouts. Invite yourself twice, mute one instance.
When I've presented at conferences, remotely, I've driven two instances of google hangouts in exactly this way. So long as you've got one of the instances' audio completely cut out (I recommend the screenshare), and enough bandwidth to push both, you're just fine.
You have to use multiple google accounts (but who doesn't have multiple accounts) set in different sessions of chrome. When I presented (and tested just now) it also requires the primary initiator to be either in a hangouts on air (you don't have to broadcast, but it turns on broadcaster tools to mute participants) or choose a moderated video call -- whatever means (and it changes every few months) to get to the control room app.
Hangouts also features google drive, google slides, and rizzoma integration, which may reduce the need for dual-channel screen sharing. You may also find a hangouts app that serves your purpose, though I've had poor experiences with most, save for drawing a beard over my beard (which was amusing.)