i have used the method you described many times. it works well. just make sure of a few things:
always handle the dry ice with gloves.
don't let the dry ice come in contact with water, it will sublimate very fast and not give you the refrigeration effect you are looking for.
cover the cooler with a blanket while you are doing the freeze for better effect.
i usually get the cold water to fill the voids by putting ice in a bucket, adding water to the top of the ice, stir it around for a minute or two and then pour it in. that will get the water to 32 degrees. you don't need to fill to the top of the blocks, just till you can see it.
be careful putting fresh veggies in that cooler. you can get that ice to maybe 30 below zero. it will freeeze beer.
play with it. after a try or two, you will be the one with ice at the end of every trip.