In the U.S. can a Illegal immigrant travel to another state by air with only his countries passport?

I hope not.

Yes, there is no immigration checks for local flights within the US, As long as their ID is valid and as long as they are not on the "watch list"

I would suggest not to...

My own experience... on December 2007, I went to Chicago for business meeting.
While waiting in line for ID check, I noticed one passenger was escorted out by two immigration officers.

During the year 2006-2007, number of deportation cases in our office increased roughly 100% and in most cases, they were arrested at the airports.

Surely your ID would be your passport, which would need to show a US visa / stamp? In which case, you could be easily caught.