He played for the national team at the World Championships in 1952, 1953, 1954, and 1959. Unsinn also played in the 1952 and the 1960 Olympics. He played for EV Füssen from 1946 to 1960 and for ESV Kaufbeuren from 1960 to 1962.
After that Unsinn was a coach for various clubs in Germany and Switzerland from 1960 to 1981. He was a national team coach from 1975 to 1977 and again from 1981 to 1990 when he retired for health reasons.
Unsinn is a member of the International Hockey Hall of Fame and the German Hockey Hall of Fame.