Yes, 2 pairs of sneakers - lesss chance of rain in the AM - so do the pool at the resort - then go to a park.
AK - bad in the rain - do not do
EPCOT, MGM STUDIOS and MK - all have indoor rides. Wear your poncho's and go. It's warm - just a bit wet.
Still raininng at night ? Kids want out of the room, not to say a cranky DH ....
go to DTD - take in a new movie if one is out, go to Rain Forest Cafe and have some fun with dinner. go to the World of Disney - biggest
Disney Store in the USA (not sure about NYC's store though) - don't want you items to get wet - either have the Disney Store ship them to your home address (less to pack) or have them sent to your resort (as long as you are still at the resort 2 days later)
If you can swing the $$'s - see if there are any tickets for the HDD - it's all inside - kid friendly food and a very funny slap stick show. Some discounts at the 9:30 PM show (maybe Disney Visa, Annual Pass, not sure of others)
Just remember you are at WDW to have fun - so smile
