Does the DNA take that long typically? They said they ran it through the State Police. Do they run full autopsies on these John Does.
1. There is a big issue with backlogs when it comes to forensic DNA testing. There's only a small number of laboratories which can handle human remains and testing is done in order of who showed up to the office first. Since his cause of death was undetermined, he unfortunately probably was not considered a high-priority case to work on in comparison to, for example, people whose deaths were ruled homicides.
Like the rest of your body, DNA immediately starts to break down the moment you die. You need a sample of DNA that is unbroken for a long enough length of material in order to use it forensically. Since he was recovered from a body of water (and I apologize if this comes across as gory), the decomposition process would have been different than a body on dry land and this may have made it harder to obtain DNA that was complete enough to be used for comparing the body to your cousin. If this was the case, special equipment and techniques would have to be used to find usable DNA and this would have added to the amount of time it took to make the connection between him.
From the things you have written I assume your cousin was reported as a missing person and that a close family member had DNA on file with the police to help find him. If he was reported missing after the body was found, they would have been unable to make a match between the length of time in-between him being found and the sample being processed because they would not have known he was missing.
2. Oftentimes if a death isn't considered suspicious an autopsy will not be performed even if the cause and manner of death (homicide, accident, suicide, natural, etc) isn't obviously apparent because the process requires a lot of paperwork and can be very time-consuming. I don't know the specific policy in Louisville, Kentucky though. I will say (and I again apologize if this comes across as gory) but the specific ways in which water damage affects decomposition can make it very hard to determine a cause of death. If you have concerns about his death possibly being suspicious, I would reach out to the medical examiner and police again and make these concerns known, especially if there was something concerning about the circumstances under which he disappeared.