Place the yellow-fleshed and sweet potatoes in a 6- to 7-qt. slow cooker. Add the chicken, broth, scallions and garlic; season with salt and pepper. Add 4 cups water and tuck the corn around the chicken. Cover and cook on low heat until the chicken is tender and an instant-read thermometer inserted into a thigh away from the bone registers 165 degrees , about 6 hours.
Transfer the chicken to a shallow bowl, leaving the slow cooker on. When the chicken is just cool enough to handle, pull the meat from the bones, discarding the bones and skin.
Skim the fat from the cooking liquid. Smash some of the potatoes in the slow cooker. Add the pulled chicken and stir in 1/4 cup cilantro. Top the soup with the avocados and sour cream. Garnish with another 1/4 cup cilantro.