Actually - you have 4 differences ... and no, I'm not counting the air.
Sugar Granule Size
: C&H doesn't say what calibration (size) their crystals are, but Dixie Crystals is a med-fine size (although it is marked extra-fine on the bag), according to their website. Since granule sizes can range from med-fine, to fine, to extra fine, to super fine ... you just have to look at the bags and try to find a brand that matches the granule size of the C&H. Of course, there is the matter of the sugar - to keep this part equal, make sure the bag says "Pure Cane Sugar" - if it doesn't it could be beet sugar, or a blend of cane and beet. This, I have read, can make a big difference in baked goods. And, as Lulu said - throw your sugar into your food processor with the steel blade and take it for a spin - until it is the consistency you are looking for.
: All-Purpose flour in CA is not the same critter as what you get in NC - they are blended a little differently - especially if you are using a "local" Southern brand. But, even national brands who claim they blend theirs the same for distribution all across the country show regional differences - for example Gold Medal flour purchased in CA did not perform the same as Gold Medal flour purchased in TN. The Artisan
website has some good information on this and the test they conducted on their Flour Test
: The pH and hardness/softness of the water can sometimes make a difference - especially if your recipe is using a chemical leavening agent like baking powder or baking soda - but it can also affect yeast leavened breads, too. It can also affect the flavor.
Now ... even if all of those were the same ...
: Different ovens behave differently. Gas ovens produce water vapor - electric ovens don't. Thermostats are not always (seldom) accurate ... if you used an oven thermometer to verify the temp of your oven in CA you can use it to get your oven in NC to the same temp ... but if you didn't ..... you're going to need to get one and do a little experimenting.
It will just take a little time experimenting to figure out what adjustments you need to make using new stuff. Making a perfect batch of cookies after moving from one side of the country to the other shouldn't be that hard, should it? But, it can be.