In general, two different kinds of Asian persimmons thrive here in the northwest, 'Fuyu' and the larger-fruited 'Hachiya.' For best eating, the fruits must be harvested when fully colored, the fruits cut from the tree, each retaining a short woody stem. Depending upon the kind at hand, storage at room temperature until fully mature may require up to a month or so.
The shape of a 'Fuyu' resembles a orangish tomato that is slightly flattened. It can be eaten when firm ripe and fully colored, or after the flesh softens a bit. Don't wait until it is mushy.
'Hachyia' are very astringent -- the extreme puckeriness you observed -- but are delicious when soft ripe. Then, they feel like a soft bag of pudding. To eat, cut the fruit in half, then spoon out the soft flesh. The fruits will slowly ripen off the tree if stored at room temperature. And, as mentioned above, to harvest, cut each fruit from the tree and retain a small amount of stem.
Freezing a 'Hachiya' fruit overnight will soften the fruit and also remove the astringency. To eat, spoon out the sweet, soft flesh while still partially frosty or wait until fully thawed.
The frozen flesh may also be used in baking. And some ethnic groups, such as the Asians, like to dry the fruit for later use.
If you plan to donate the persimmons to a Food Bank, etc., understand that it can be difficult to give them away! That said, Asian people/groups may more than welcome such a gift.
In any case, be certain to call ahead to determine if the agency/group is interested and how many they would like to receive. They may not have storage space or they don't have clients who enjoy persimmons.