Although I have not been using Limewire that long, it has never saved the results of my searches. However, in order to save your incomplete downloads in the action box ready for subsequent reconnection, you have to close Limewire through Quit. If this is not happening, it suggests that you have a malfunction in your copy of Limewire. You should uninstall and, using Finder, trash everything with "Limew" in the label. Then download a new copy and install that.

However, watch out for the Kill Inactive button. That will inconveniently remove incomplete files awaiting reconnection as well as those that are complete. Use Kill Download to keep your download window in order.

And in general, if Limewire crashes, it will always lose the incomplete files from the action box and nothing that you might tweak will prevent that.