Sarah,
I know what you are going through. I had a relative pass away who was a bit of a hoarder when it can to plastic models. I bought a glass display case and put maybe 25-30 on display in my house. Problem is I had to rent a storage unit to store the thousands that he had collected over the years. They are all very nice models just sitting boxed up. I have given away many over the years. Some to children, some to veterans. I know what you mean about not being able to throw them in a dumpster. Good luck with finding new homes for the ships. I have enjoyed giving the models away to people that have really enjoyed them. It may just take a little time.
Duane