PWP Do you have a green card or US Passport? If not, are you aware you cannot work there; you'll be able to do some work experience as a flying instructor following your FAA courses (assuming work available, but don't expect to be paid much), but you'll be on a flight home when your student visa expires? You'll then need to do a JAA licence conversion. That's now simpler and less expensive than it was, but you'll still have to do all the JAA groundschool.
Some airlines here are not so keen on this route; many people have to spend a little time working their way 'up' to an airline job here so it can be quite a slow route if you are in a rush for that 737 job. On the other hand, if you don't have commitments, cannot get sponsorship and have limited cash, it's requires less capital than some other routes.