If you don't want to use email, and if your old machine has a USB socket, have you considered a memory stick? Ebook files tend to be quite small and modern sticks can have a capacity of 8 Gb or even more. They're getting much cheaper, too, and are generally useful things to have.
If you are a Mac user you may be able to Firewire the files over; otherwise you can set up a network using an ethernet cable, but that's all rather nerdy and beyond the likes of me.
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