I put some pork shoulder in the slow cooker along with water, lime juice, apple cider vinegar, and spices. I then cooked it until it was falling apart. There's enough to last about a week.
Should the leftover meat be stored in the juices, or will the acids break it down too much? Would it be better to store the liquid in a separate container, and just spoon some over the meat before reheating?
If not consuming right away, it is often recommended to allow meat that was braised to cool, and be refrigerated in its juices. For this reason, many recipes suggest making a braise a day ahead for better flavor. I don't think your concern is break down of the protein, as much as it is shelf life. If you are not going to fully consume the product within 3 - 4 days, freeze a portion of it.
Your braising liquid is now a delicious stock. Congrats! You certainly can store the meat and stock together, but the meat has no color and the result is too wet I find. The meat still has plenty of fat to be succulent on its own. I reserve the stock in the fridge and separate the lard. With the remaining stock I cook black beans. When I reheat the pork, I reverse sear the pork in the reserved lard. The meat simply won't dry out, it's too fatty.
I usually keep a part of the meat in the juices and reheat it that way the next day. Sometimes I divide the meat and liquid and have one portion in the fridge and one in the freezer.
If I have enough meat, I will take some out of the juices and cool and/or freeze separately. Depending on the meat it can be used as is or has to go in a dish with a lot of liquid.
You can see from that I feel you can do both, in and out of the juices and fridge as well as freezer.
