Question

This is not quite an answer to your question, but you might consult the book by Donaldson and Kronheimer "The geometry of 4-manifolds". In chapter 2 they prove an integrability theorem for holomorphic vector bundles, the point being that this can be regarded as a simpler version of the Newlander-Nirenberg theorem, and (in my view) very suitable for your course. You might also want to mention the following simple example for instructional purposes: the nilpotent Lie group H^3 x R where H^3 is the Heisenberg group has an obvious left-invariant almost-complex structure whose Nijenhius tensor vanishes. Although not a complex Lie group, it is easy to find independent local complex coordinates z_1, z_2. I suspect that there are similar classes of almost-complex examples where the integration is elementary.