The only Robert Moore in the Tampa Tribune index with a death year of 1978 was 37 years old at the time of death, and the death was in a hospital. Obviously, this isn't the Robert Moore you're looking for.
I'm not sure if your Robert Moore would have been included in the index if there was only an article about the death, and no obituary, but I have my doubts that an article would have been indexed.
The next step would be to get the death certificate from the state of Florida.You can find the department of vital statistics website by doing a search of the internet.Certificates are only $5.00 and it will give you the exact date of death, or date the deceased was found. I doubt if you'll get a librarian or volunteer to search over 30 days of newspapers with out an exact date of death.