Nobody can guarantee the health of any animal. They have behavior problems, that's why they were given up. You will find many pets abandoned in shelters are not give up due to behavior problems, but due to pet owners not having the time or energy to train and redirect behavior problems when they arise. The majority of shelters and rescues have a staff of volunteers and foster families dedicated to redirecting improper behavior problems prior to adoption. The majority are strays that were picked up by animal control or rescuers.
All shelters have are adult dogs and cats.
This isn't true, kittens and puppies go very quickly in shelters. Give some thought to the adult adoption. More healthy adult dogs and cats are put to sleep in shelters then puppies and kittens. These adult animals are not a surprise, what you see is what you get. You will already know how big the animal is, they already know basic commands, and do not need the constant supervision and training that goes along with owning a kitten or puppy.