You may be able to, but I wouldn't recommend it.
The TSA website lists the types of identification that can be used for domestic flights - and the only one you're likely to own is your passport.
If you do not have acceptable identification then that doesn't immediately mean you can't fly, but it will at best mean additional time in security whilst they attempt to confirm your identity. Given you are a foreigner they may or may not be able to suitably do that, which means that you may or may not be allowed to fly.
This will be in addition to any requirements your airline has around identification - especially if you are checking bags they often require it.
And then even if you are allowed fly you are going to find yourself a long way from your passport. Although it's not generally a requirement for foriegn nationals to carry their passport at all times, I certainly wouldn't recommend being too far away from it - especially given you are from South America, and you're in what is somewhat a "border" state.