And just like Abbas' family, his family fled way before any Jewish troops entered the town. Just like Abbas' family, they left because they were afraid Jews would act like Arabs and take revenge on the Arabs for their massacres of Jews in 1929.
Here's is the Palestine Post article about Arabs fleeing Safed while the Jewish army was still five kilometers away:
This Arab flight mentality was noted the day after, and it was in no small part due to the fact that Arab leaders would flee first, abandoning their people:
The Haganah did end up entering Tzfat a few days later where they encountered heavy resistance from Syrian and Iraqi troops before taking over the town on May 11th.