The dry ice will "melt" first (vaporize) at around -109°F and your regular ice will not melt until it reaches 32°F. That means the dry ice will vaporize first, then when the ice finally warms up to 32°F it will begin to melt. That is why the dry ice was gone but not the ice.
Since most ice is only around 30°F to say -10°F, when the dry ice is put with it, the dry ice begins to vaporize as is drops the temperature of the ice. That is why the recommendations to get really cold ice is good since it won't put that demand on the dry ice.
Block ice requires more heat to melt than the same volume of ice cubes, that is why the idea of block ice is good. Ice cubes have a lot of air in between, and the air does almost no good since it takes very little heat added to warm up air. Far more heat is required to heat the same volume of ice/water than the same volume of air.
Just for the heck of it, you might want to Google the "heat of fusion of water". It is an interesting engineering concept that helps you understand that considerable amount of heat needs to get into your cooler to turn 32°F ice into 32°F water. The more you slow that heat transfer down the better.
The better your cooler the longer your ice should keep. When you consider the price you paid for your ice ($80+$10), a premium cooler begins to make sense after only a few trips if you can cut that dry ice usage down. The sleeping bag ideas are good ones for long term ice storage where you don't have to access the ice easily.
One more thing, think about ventilation and shade for where your ice chest is kept. You know how hot an enclosed space (like the truck cab) can get when the sun is beating down on it.
As Tim mentioned putting insulation under the bottom is a good one. There are reports of even some of the high end coolers that have marginal bottom insulation. If you think about it, placing the relatively smooth bottom of a mid grade ice chest on say on your metal truck bed is kind of like putting a pan on a smooth top stove, you have a very efficient path for heat to conduct into your ice chest. Conduction is a very efficient way of heating.