Unless your fridge is above 40 F (4 C), it won't go bad. I've had large frozen turkeys take 4 days to thaw before. Just use it once it is fully thawed, don't let it sit around thawed for several more days.
There's no food that I'm aware of that goes bad in the time it takes to thaw it. The stilltasty link below shows 3-5 days for refrigerated meat, and 4-6 months for frozen. Transitioning from frozen to thawed though, I generally count on defrost time + 2-3 days.
If you want to thaw it faster you can always use the cold water method. Simply place it in a sink full of cold water (submerged) and change the water every 30 minutes. The rule of thumb for turkey is 30 minutes of thaw time per pound. I imagine pork shoulder would be similar, maybe a little longer?
Additional Info:
http://stilltasty.com/fooditems/index/18060
A Southerner will probably tell you that there is no type of BBQ sauce to go with this type of pulled pork because it's not actually real pulled pork... or real BBQ. Authentic pulled pork is smoked, and that comes with a completely different set of pairings. This is technically just braised, shredded pork.
Which is not meant as a criticism, mind you - I often make a similar recipe in the winter time using a dutch oven. But terminology is important here because there's a huge difference in flavour between pulled pork and braised pork. I'll note for the record that any kind of slow-cooking will make a pork shoulder fall right apart; you do not need the root beer (which isn't that acidic anyway, and acid-tenderizing is somewhat of a myth) to tenderize the pork. The heat and moisture do that all by themselves, although root beer is as good a choice as any if you're braising.
Now onto the flavouring. Because the braise, unlike smoke, only minimally flavours the meat, you're going to need strong flavours from elsewhere. If you're really trying to mimic an authentic Southern-style pulled pork then you need to incorporate the smoky flavour somehow, which means using a smoky BBQ sauce (hickory is common) or better yet, incorporate this flavour at the very beginning of the cooking process by applying a dry rub containing smoked paprika or including a small amount of liquid smoke in the braise.
If you do use a rub then the other flavours are up to you - there are plenty of recipes out there - just don't use salt in it because pork already has plenty. As with any braise, stick to aromatics like garlic/onion powder, various chili powders/peppers, cinnamon, etc.
Once you've successfully incorporated the smoke (or what passes for smoke in indoor cooking) then a little sauce goes a long way. Definitely incorporate some of the leftover braising liquid so as not to waste those delicious drippings - I use about half of it but that's with a much milder braise. Ketchup and dijon mustard in a 50/50 to 60/40 ratio is very simple and surprisingly effective when mixed in with the juices.
If you still find this to be lacking punch, you can throw in some cider vinegar, which pairs well with both the pork shoulder and your root beer braise. I actually boil off the pan (after reserving some braising liquid) and use cider vinegar as a deglazer for the fond, but since you're using a crock pot (or equivalent), you probably won't have much/any fond and this would require an extra vessel, so... just throw a splash or two into the finished sauce, and reduce if that makes it a bit watery. Worcestershire sauce is also a good way to punch up any homemade sauce, and if it starts to get too intense or tangy from either that or the vinegar, tone it down with some brown sugar or molasses if you have any.
If you're dead-set on following the recipe to the letter, then, as above, I think I'd choose a hickory or mesquite sauce, something smoky and not overly thick. Some of the better brands/products will actually include a small amount of alcohol, like whisky or bourbon. Almost all will contain onion, garlic, paprika, sugar, molasses, and vinegar (often cider vinegar, as above) - not surprisingly, exactly the same kinds of ingredients as I'm suggesting in the dry rub and homemade sauce.
Best Answer
You could probably get away with pork shoulder, although you will want to consider that the texture may not be what you want -- it may either have too much fat and connective tissue, or it may fall apart too much, depending on how long you braise it for. You will probably have to alter your cooking times, I'd imagine.
Pork hocks are somewhat similar in texture to the shank, so that a possibility. Though you will probably run into the same availability problems.
It may take a small amount of modification, but your best bet may be to go with country-style ribs. They are tender and flavorful when braised, and are readily available in any grocery store.