Which one of these chordate groups lacks a post-anal tail and a notochord as adults?

The answer is tunicates.

Tunicates belong to phylum Chordata. Thus, it is expected to have dorsal nerve cords and notochords. But, as adults, they lack a post-anal tail and the notochord. The reason they belong to phylum Chordata is that their larval stages have all the characteristics of phylum Chordata - nerve cords, post-anal tails, and notochord. During metamorphosis to adults, tunicates lose these characteristics.