1. The teachers are busy correcting papers.
2. The teachers are busy in correcting papers.
3. The teachers are busy with correcting papers.
#1 is correct. Are #2 and #3 also acceptable?
#3 is less bad than #2 because of the preposition used, but you don't need a preposition immediately before an ...ing verb form.
#1 is all you need.
'The teachers are busy with their paper correcting.', where "correcting" is a noun, would be correct. But it is simpler and more natural to say #1.