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Yes, it can make a noticeable difference.  Having lots of wet surface area was one way rooms were cooled before air conditioning.

There is an example (at least used to be) of this at the Smithsonian Museum where water is pumped to overflow a large urn, with the outside of the urn shaped to maximize surface area of the water sheeting down its sides.  The room is noticeably cooler than adjacent rooms.

One problem with your laundry method is that the amount of water is limited.  Of course you also need the ambient humidity to be low, else there won't be any evaporation, and therefore cooling.