In my opinion, the trick is to not use a slow cooker for chicken at all. The dish you are describing (kind of simple tom ka gai, it seems like) would normally be built quickly in a wok. Here is how I would make a dish like that on a busy day:
Steam your carrots (or celery or broccoli or cauliflower if you want it) for about two minutes and then dump the hot water out and run cold water over them to stop the cooking. They will be just a little soft on the outside. You can do this the same day you buy them. The same day you buy your chicken, reserve the portion for the soup, chop it into half or three quarter inch cubes, salt and pepper it, and put it in a baggie in the fridge. Chop your onions, grate your ginger, etc. The day you want soup, if all of this is done, you will be able to prepare an amazing soup in less than 15 minutes. The key is do all the preparation beforehand.
To make the soup, lay everything out. This includes the utensils and even the serving dishes. When you are ready, heat your wok and add oil. Let the oil get hot and add garlic. About 20 seconds later, add the chicken. Let the chicken get nice and brown and toss in the onions and carrots. Let them get a little brown (keep everything moving by the way) and add any strong flavors like vinegar, mirin, fish sauce, etc. Let the meat and veggies pan braise in the liquid. At this point, I just reach in and grab the biggest piece of chicken I see and slice it in half to gauge the cooking. You want it to be just barely undercooked - this means no raw color, but a little less cooked than you want to eat. Add the coconut milk and the lime juice at that point. Taste the broth as it cooks and add salt, fish sauce, or lime juice until it tastes perfect. When the coconut milk is hot enough to serve, you are ready to eat. Add your cilantro at that point and maybe a sliced chili pepper if you want a little spice.
Again, just do the prep work the night before and it will only take a few minutes. As you make the recipe more, look for efficiencies. You can season your coconut milk and put it into a cleaned mayonnaise jar so you just shake it and dump it and don't have to fumble with the can opener-- that kind of thing.
This answer is turning into a novel, but I want to answer the question about beef as well. Beef is good for the slow cooker, but you want to get tough cuts of beef with big chunks. Think butt roast, shoulder roast, etc. The names in grocery stores can be really different, so the best strategy is to ask the butcher for a good cut for slow cooking. I like to cut the beef into two to three inch cubes-- really big pieces.
There is not consensus among chefs as to the best way to keep the meat tender. Some recommend cooking the outside to create a seal. Whether it makes a seal or not is up for debate. I personally do it this way. I either dredge it in flour and brown it in oil, or salt it heavily and pan sear it. Other sources recommend putting the raw meat into the cooking liquid at various times, either into cold or hot liquid. You will just have to experiment and find what you like. Six to eight hours cooking in liquid is a really long time, so you want to do what you can to not overcook it.
When done right, braising, or any other low temperature cooking is fantastic, but when any meat (or poultry) is overcooked in liquid, the tasty fats seep out and plain water seeps in. Once that has happened, it is the same thing as soaking an overcooked steak in broth; it is wet; not moist, tender and flavorful.