About giving to tzedaka to those who don’t deserve it: I heard this on the words “Hashem tzilcha.” If we give to someone who may not deserve it, but we’re not sure, Hashem will give to us (like a shadow following our actions) eve if we don’t deserve.
I would urge everyone to contact people in their communities to have the kochos to set up a vaad. Then when a collector comes, if s/he has a vaad letter, you can feel comfortable writing a check or giving freely, and you can give small cash donations to those who don’t if you feel uneasy.
As far as the original topic, what to do about so many, that is difficult. Ask a rav how to triage so you can do so in good conscience. Say the tefilla in Shema Koleinu, and read Birchas Haman daily to thank Hashem for letting you be in the position of being able to give, and to daven for those that they should have kol mishalos libhen filled l’tova.