I'm sorry, but I fail to see how you were obstructing governmental procedure by providing them with the information they requested. What if you couldn't speak due to some sort of disability? Would they charge a deaf man with this same crime? Sounds a little ridiculous.
I had a similar case in Gainesville, Florida. I was stopped for "walking on the right side of the road" at 2 or 3 in the morning by some a-hole cop on a power trip. He explained why he stopped me, so I told him that I was not from Gainesville and had no idea that I couldn't walk on the right side of the road. I told him that I would gladly cross the road at the next legal opportunity to do so, and continue my walk home from work on the left side of the road. The officer asked me where I was from, and I told him that I was from Denver, Colorado (I'm actually from Manchester, New Hampshire, but I had a warrant for my arrest over non-payment of fines from New Hampshire, so I wasn't too comfortable telling a police officer that I was from NH). I told him I did't have any form of identification on me, and gave him a false name. The fine gentleman told me he was going to call Colorado or something like that and ask if they had any record of the (fake) name I gave him ever having had a state-issued identification card. I was pretty tired at that point and didn't feel like walking anymore, so I told him the truth and provided my identification.
I was immediately arrested for "providing false information to a law enforcement officer" or something like that. It was subsequently found that I had an outstanding warrant from NH, and I was denied bail for the night, pending NH's decision on whether or not they wanted to expedite me from FL over an $800 fine (which was originally a public defender's fee from a non-DUI driving offense: habitual offender driving after suspension; hence why I was walking). Shockingly, NH didn't feel like spending thousands of dollars to expedite a non-felon (habitual offender is actually a felony in NH, but I got the charges dropped without the help of my public defender, which is why I chose not to pay for his services) from FL over an $800 public defender's fee, and they dropped the warrant.
By the way, if you want to get a warrant for a petty offense dropped, this is a great way to do that.
Anyway, the false info charge still stuck, and my public defender in Florida was more than happy to have recieved my case. He was very adamant about fighting my case in court, as the local police had actually been charging people all over town with that same charge, most of the time without standing. He told me that it didn't apply in my case, and knew that we could win. He was actually annoying me, because I already knew that after researching the laws on my own.
I ended up moving back to NH, seeing as how I no longer had any warrants from there, before the trial. The public defender actually found me and tried to get me to try the case via television (lol), but I declined. Florida actually dropped the charge when I got picked up on the warrant for failure to appear while I was in California (yeah, I travel a lot).
Don't worry so much. Just because you are charged with a crime doesn't mean you'll get convicted of it. That's the beauty of the judicial system. Cops will charge anybody with anything if they're having a bad day, and they usually just rely on people being stupid enough to think that they can't beat the charge. This seems like that kind of case. If I were in your shoes, I wouldn't even apply for a public defender, but you'd probably be better off doing so if you're not familiar with court procedures.
Good luck, man. Hope this helps, somehow.