Oatts' work on woodwind instruments (saxophone, clarinet, flute) became widely known when he led the crossover jazz group Flim & the BB's in the 1980s and '90s with bassist Jimmy "Flim" Johnson, drummer Bill Berg, and keyboardist Billy Barber. Since the 1970s, Oatts has released more than a dozen albums as a leader and co-leader, and he has appeared on over 100 albums as a sideman with Joe Henderson, Jerry Bergonzi, Eddie Gómez, Bob Brookmeyer, the Vanguard Jazz Orchestra, Joe Lovano and others. He has also accompanied Joe Williams, Sarah Vaughan, Mel Tormé, and Ella Fitzgerald.
Oatts has taught at the Manhattan School of Music and has been artist-in-residence at the Amsterdam Conservatory. In 2006, he became a professor and artistic director of the jazz studies department at Temple University in Philadelphia.