You need not case this sausage. I often make it loose, as most Mexican recipes call for uncased chorizo anyway. In this case, there is no need to let the finished sausage rest overnight. Many Mexican chorizo recipes use vinegar, too, but this damages the sausage's ability to bind to itself. Use vinegar (any kind) instead of ice water, if you are not casing your links.
Take out some hog casings and set in a bowl of very warm water. Mix the ice water with the achiote paste and chill it in the fridge.
Grind the meat mixture through your meat grinder (you can use a food processor in a pinch, but you will not get a fine texture), using the fine die, about 4.5 mm. If your room is warm, set the bowl for the ground meat into another bowl of ice to keep it cold. Make sure the meat mixture is very cold before moving on to the next step: You want it between 27°F and 35°F.
Add the water-achiote mixture and mix thoroughly either using a Kitchenaid on low for 60 to 90 seconds or with your (very clean) hands. This is important to get the sausage to bind properly. Once it is mixed well, put it back in the fridge and clean up.
Hang the sausages in a cool place for up to overnight (the colder it is, the longer you can hang them). If it is warm out, hang for just one hour. Once they have dried a bit, put in the fridge until needed. They will keep for at least a week in the fridge. If you are freezing the sausages, wait a day before doing so. This will tighten up the sausages and help them keep their shape in the deep-freeze.