"I'm looking to get into grass-fed/pasture-raised/natural etc. meat for my family"
This takes a bit of work, research and traveling on your part. One processor will probably not be able to supply what you want and probably will have no idea how raised. Research, then call each specific farm for each type of animal(s) you want, make an appointment and then get in you car and go take a look at the farm. Wear boots and take a large garbage bag to put those boots into and shoes to drive home in. If you see what you like, the farmer will give you price and where he can deliver for processing. Then contact the processor to discuss cutting and if possible go have a face to face. Usually these places have standard packages that they do and if you want non standard, you have to be very specific about what you want and how you want it. You then pick up what you want in large coolers and its already frozen. Small chickens may be processed by the farmer and you usually get a chicken or any fowl fresh in a large heavy duty poly bag. I don't purchase frozen from any farmer because you just don't know how long they have been frozen.
It's well worth the effort but be aware that you may or may not end up with meat that you like for two reasons, first is that animals vary as to how they Grade out and if you want USDA choice, it may not be your idea of USDA Choice and seondly, you are used to eating supermarket beef, pork and chickens. Chickens particularly if free range are different in that the joints are much harder to separate but you won't get a waterlogged bird as in most supermarket chickens.
Frankly for beef, our local Pick n Save has great USDA Choice and when on sale I purchase Primal cuts and process myself. Pork is one area i haven't found what I want, as market hogs are bred and raised to be very lean and I know of no one in this area selling the fattier breeds.
I have a farmer that does excellent birds of all types but its seasonal and you must buy what you want and freeze until the next lot is ready.
Good luck.-Dick