KDWH, also a class D, is where I've done my training. My first two flight instructors taught me to contact CD. It's usually the same controller working CD and ground, and sometimes they won't even respond to you on CD for a VFR clearance, but I always tried calling them their first. Most of the time they'd say "roger, contact ground on 121.8 blah blah blah" and then I'd receive my official clearance into Houston Bravo with ground after I had begun my taxi.
The last flight instructor I had told me it was unnecessary to contact CD for VFR at our field, and I noticed that if I called ground and skipped CD they seemed indifferent; well, at least no one ever barked at me. Toured the tower a few months after my commercial and asked the tower manager what they preferred up there and he said just to contact ground for VFR.
So, it's probably good practice at an unfamilar airport to try CD first, and to definitely do so if ATIS instructs (obviously), but overall I guess it's just an airport to airport thing based on their operations, etc.