Tunes such as “Mr. Jones” and “Accidentally in Love” have helped Counting Crows sell more than 12 million albums in the U.S. But Adam Duritz wanted more than a hit record or a few songs on the radio.

For every person who says, “Counting Crows, are they still around?” is a legion of fans still hungry to hear one of the biggest pop bands of the 1990s hit makers, which burst on the scene in 1993 with ...