**Edit: This answer was wrong - I misremembered the result. See Allen Hatcher's answer. Of course, the references in the second paragraph still stand.**

Those two operations give trivial outer automorphisms of $\pi_1(B)$, whereas the handlebody group surjects $Out(\pi_1(B))$. So Dehn twists about disks and annuli don't suffice (in fact, I think it's known that these generate the kernel of the map to $Out(\pi_1(B))$). 


Suzuki [gave generators][1] for the mapping class group of a handlebody, and Wajnryb [gave a presentation.][2]  See also [Popescu.][3]


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=433433
  [2]: https://eudml.org/doc/212312
  [3]: http://www.jstor.org/stable/43679205