I have a British passport, been living in the US for 10 months, have applied for my green card, can I travel within the US on my British passport?
In general, the only officials who care much about your immigration status in the US are immigration officials. If you travel by inter-city bus in the US, you're unlikely to encounter any immigration officials unless you're going through the southern parts of the states bordering Mexico. There, immigration sets up checkpoints on the road (I imagine they probably check passengers on buses, too).
As an English-speaking white non-US-citizen, whenever I encountered one of those checkpoints they simply asked what my citizenship was (they ask everybody that question), and waved me on without actually asking to see my passport. But if you have your passport with you and you're in the country legally, you won't have any problems even if they do ask to see your identification.
Yes, according to this TSA website your "Foreign government-issued passport" is about the only thing that will get you on a plane. If you want to travel via car or bus, you don't even need your passport, although you might need it on a train. The U.S. is the U.S. (i.e. it's one country with 50 united states, so feel free to look around, from sea to shining sea!)