That is new information, thank you for providing it.
This unfortunately leaves only the possibility of an internal hardware failure. This means that the tuner card/chip inside the tv has failed.
This problem is very common with these units. These RCA tvs are at the very bottom of the quality spectrum, they are built with the cheapest possible components, and have very short service lives. Especially for portable equipment, it is rare for these devices to last more than a year or two, which is really all they are designed for. They are essentially built to be disposable.
Fixing this situation would not be feasible - rca provides no support for their products in any form, there are no parts available, and labor costs would be easily double the value of the unit. You could replace it with a brand new one for under $75, which is really the only feasible option here: http://amzn.to/1QKF62L
You could also connect it to an external tuner to replace the failed internal part, if it will be stationary. Those are about $30, but require power and would not be suitable for portable use. If you'd like more info on that option, let me know.
Best regards, ***** ***** for the news,