I can't say that I've ever seen 5.56 dies, but I've loaded thousands of .223. For me, it doesn't matter if the brass is .223 or 5.56 when I'm reloading. I always use .223 load data and I never load to max load.
The only time I have any concern about the differences is if I'm using actual factory loaded 5.56 ammo.
First Bolded part is true.
Second bolded part.....I always search for the max load that is most accurate.