First off, I'm not familiar with agencies and their contracts, so
you should consult a lawyer about that.
The H1B visa is transferable between companies. A company willing to take you on with an H1B can go through the required legal hoops and have it transferred to them. A Masters degree is helpful, but not hugely so. Already living in the US, having a good grasp of English, and of course being very good at what you do, are assets that are at least as helpful, and probably more so.