To go off of this, inflate one of your punctured tubes. If the hole air is escaping from is on the inside of the tube (side where nipple is on), check your rim. If it's on the outside, check your tire for debris.

Look at the sidewalls of your tire and see if any part of the general shape of it is flattened. Replace the tire if you think it needs to be.

I hear armadillo's are good but are way heavy. I personally ride in Tucson, AZ with gatorskins and regularly run over thorns, putholes (Tucson roads are...the worst I have ever witnessed) and have had no problems with them. Gatorskins are essentially the unofficial tire of southern Arizona.