No expert here, but it looks like no diseases I've ever seen. I would take a sample to your local extension office - or email photos, if in-person isn't possible - for expert opinion relating to your area.

http://sfyl.ifas.ufl.edu/map/index.shtml

Wondering whether this could be damage stemming from recent hurricane?

That's what I would say--it looks more like mechanical damage than any disease/fungus, and it only occurs on the parts of the fruit that could have been banging against something in the storm.

I pulled a few off. It’s a crusty sticky substance. This has all occurred in the last couple of days. The storm was over a month ago. We picked 2 within the last week that were fine and no signs of this anywhere. Will go to extension office tomorrow. Thanks for your responses

The stuff around the edges of those wounds really looks like sap - which bleeds from injuries to papaya fruit's skin. if it's whitish/clearish & somewhat rubbery, I'd venture to say that's what it is. An animal or chewing insects could also be the culprit. could something perhaps been scrambling up or down over the fruit? Or might someone/something have been pulling on those branches (or trying to pick the fruit) & scraped the fruits?

how about our friends, the squirrels?