Question
why does the new growth when it reaches full growth turn brown and fall off?

Answer
Several things can cause this condition in sago palms.  Cold injury is the first thought that comes to mind.  I have to cut off the old growth each year on mine because the old growth is straw colored and doesnt look good.  For new growth, the problem could be an insect called a scale.  Look at the fronds carefully and see if you see any bumpy looking things on the petioles and leaflets. If you do, then it is scale insects which need to be treated with an insecticide.  If you see this, purchase Bayer Advanced Tree and Shrub Insecticide from one of the big box stores.  This product is mixed with water and poured on the soil around the base of the plant.  The product is taken up systemically and will kill  the bugs within a week or so.  It also provides up to a year of protection from reinfestation.  Good Stuff.  Let me know your findings.  Thanks.