You should only need to have a government-issue photo ID to get through TSA for domestic travel (ie. driver's license, passport (any country), etc). But only for domestic travel. If you attempt to use a non-US passport (either for exit or entry to the US), then you run the risk that you are entering or exiting the USA under a false pretense. This could result in inability to travel on that trip that you are doing it, or on future travel.
It's best to have your US passport available and up to date at all times. Most processing centers have expedited service (for a fee) that should help. There are a lot of third-party agents offering to do this, but from personal experience, I would avoid them. I've had situations where an expired passport was only discovered the day of travel with my family and I had to send them ahead and drive to a processing center (Tucson) and get it done. But I did manage to get it same day, pleading my case, and although it cost me dearly with airline re-ticketing, I was able to board my international flight about a week later and meet my family at the international destination.
From personal experience, an expired US passport should be avoided where possible. It's not that big a deal to renew and you should do this at least 6 months before any international departure. Note that some airlines or airport exit points will not let you board a flight either to the US or from the US if you have less than 6 months expiration on your passport, so do this within 12 months of its expiration to avoid any issues.