As mentioned in the query, many differing details exist on Hanumad Jayanthi. Similarly on Garuda Jayanthi as well – some say it is in the month of Adi and others say it is during the months of Tai and Maci. Follow the practices of the respective places where they are celebrated. We can also follow the practices of many places where it is celebrated based on the Srivaishnava tradition. As our ancestors have not maintained a particular day in their practice, we are unable to be certain. We can certainly have faith and follow the details based on the panchangam or particular practice.
In our tradition, we do not fast or perform special Tiruvaradhanam for Hanumad Jayanthi or Garuda Jayanthi. But there is no harm in performing a special Tiruvaradhanam. Fasting is not specified. Follow the practices and tradition of the respective places.