This is a serious issue
I think this is a very serious issue. One cannot 'almost' live in this country without a valid License. It is a very basic and primary document for ID to everything.
You cannot be expected to go to work or anywhere else without driving as it is the public transport system sucks here. India is much better in that area.
My 2 cents is to bring this issue up to the concerned governer. The name and office numbers can be found from the DMV websites. Write a letter to the explaining the problems an incoviences faced. You will get a response. If by any chance you do not the... do go see an attorney. One solution to every problem in this country is an attorney. They will guide you as well as help you - if possible and necessary.
Also, it is better to bring things like this to authorities attention, instead of circumventing it by other means - like carrying a license from other states or showing an address of a friend. Who know tomorrow that small incident could bite us. They might say that we were not truthful... and God know what else... no matter whose fault it was.
Remember, they can track you down to the day, if something goes wrong and they start investigating. Tax filing, License, vehicle registered, your mailing address, you address on credit cards... banks... etc. etc... are all public records and anything and everything might work for you or against you.
Just my thoughts....
PS: I had an incident where a state tracked me down after 5 years for some local tax. Only because, I did not return my registration plates when I sold my car... and moved to another state. Both happened almost at the same time and I did not know any better.(ie.. I had to return the plates - I was new to the country and nobody told me either) I finally paid the outstanding amout with penalty and 5 yrs worth of interest.