Audio via the official MS stereo headset adapter is considerably quieter since the latest...

As seen on MS Official Site for the Adapter:

A headset with chat audio provided by a 3.5-mm audio cable and game audio provided by a separate cable with RCA or optical connectors. (Only chat audio
will be enabled through the Stereo Headset Adapter. For RCA/optical connectors [game audio], follow the manufacturer’s instructions to connect to the console.)

http://support.xbox.com/en-US/xbox-one/accessories/stereo-headset-adapter

I actually got the MS brand Stereo Headset with adapter. Its a really nice headset, I just hope they fix the audio codec issue soon.

In mine they have been selling the official stereo headset since wednesday, but no sign of just the adapter

This is correct. Using the Xbox One Stereo Adapter (support.xbox.com/.../stereo-headset-adapter), you can use any of the following types of headsets:

◦A single 3.5-mm audio cable headset.

◦A dual 3.5-mm audio cable headset with 3.5-mm CTIA audio cables.

◦A headset with chat audio provided by a 3.5-mm audio cable and game audio provided by a separate cable with RCA or optical connectors. (Only chat audio will be enabled through the Stereo Headset Adapter. For RCA/optical connectors [game audio], follow the
manufacturer’s instructions to connect to the console.)

◦An Xbox 360 Wired Headset with a 2.5-mm audio cable. Connect to the Stereo Headset Adapter by using the 2.5-mm-to-3.5mm adapter that’s included with the Xbox One Stereo Headset Adapter.

As far as suggestions, that is a little harder for me to do. I am a huge gamer on console and PC and I strongly recommend Steelseries headsets, however, if you are an audiophile or use them for more multimedia related things, someone may be able to suggest
a better headset/headphones.

Basically because they wanted to offer all audio through the controller. Game and audio chat using the 3.5mm TRRS jack so you could plug in the official MS headset, third party headset, or any old stereo headphone and have wireless audio The cost of a regular
controller would have gone up if all the extra circuits inside the chat adapter were built into every controller.

I originally just bought the chat adapter to use with my headsets AXPro and then Astro A50 I had bought to use with my 360 and PS3, but I've also used it plenty of times to plug in my earbuds so I can mute my TV while I play a game or watch a movie. Also
makes a nice incentive if you want a cheap wireless headset I've seen the official MS stereo headset which comes with the chat adapter for $50 which is a decent starting price compared to $100 or even $300 for others.