Providing practical, down-to-earth advice for family historians since 1995, online since 1985.
When NOT to find a cousin
Your cousin's former wife apparently doesn't want much to do with his family or she would have kept in touch. How many divorced women of her generation did? Divorce typically brings rifts between families, with effects running through several generations. She may have married again, and that is another reason for her dropping your side of the family. Cousin or not, please remember:
Genealogists seek to understand "WHO ARE THE PARENTS?" to quote George K. Schweitzer.
Working on a dead end descendency line is better saved for the time when the US census records for 1940-1980 are available to the public. It isn't like there is someone who could have inherited the family bible to give you advice as to "WHO ARE THE PARENTS?"
If I were the funeral home director, I would probably have responded in a like manner. There are all sorts of bogus firms offering to look for "possible heirs" and "missing family members," etc. Your story, as explained, probably didn't sound like a legit genealogical query. If there was no death notice in the newspaper for a death that occurred in 1983, that is most odd. How you ever found the name of the funeral home, I simply don't know. Turning up the heat flashing "Freedom of Information Act" will probably only raise the ire of the funeral home director.
Curiosity killed the cat.
Go forward on a line that has more potential to explain YOUR direct ancestry, not your elderly cousin's x-wife and possible descendants. We don't want to give the poor woman a heart attack.
Happy family tree climbing!