Well, I'm currently in the US on a J-1 visa and working on my instructor ratings so I do have first had experience of this visa. My school has quite a few instructors who are good bets for airline jobs in the next few months. However, if they do not leave I may need to find a job elsewhere. The official line from the school (who have otherwise been good) is that they will not permit me to find work elsewhere. Now, things are going well here in the States with plenty of jobs to chase so finding work is most likely not a problem; I may well be able to find something much better that what my trainer could offer.
I sense that the story I'm getting from the school I'm training with is not true. The INS is not easy to contact so if anyone can offer me any advice I'd be very happy.