My work-around for the past few hours has been to disable the Bug Reports by setting it to "Always Discard Bugs."
That seems to at least provide a temporary solution; I can't say if it will last, but if that helps anyone else, it has helped me maintain a connection for a few hours now.
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Edit: I also upgraded Limewire to Limewire Pro 4.9.4. I still have to disable Bug Reports and the OS X skin is buggy, but hopefully having a newer version will be better all around.