To figure out which shelters have younger animals available, try using a service such as PetFinder.com to seek out a specific age group. Kittens and puppies can generally be adopted out once they are 8 weeks old. Therefore, that could be a good starting point.
Not all shelters and rescues will list the younger animals, as those can tend to go faster than allows time for a website listing. Because of this, I recommend visiting in person if you are having trouble finding information on younger pets online. This could also be one of the reasons this myth gets spread.
If you are interested in tougher cases, there are often mama cats and dogs with their offspring in the backrooms of shelters and rescues because they cannot be adopted out and do not have a foster home. These animals are considered special needs and may require a temporary foster period and rehome of some or all of the babies before final adoption of the mother.
Remember that just because an animal is not listed online does not mean that animal does not exist within a shelter or rescue facility.