Way to Copy Text From OS X to Evernote and Keep Formatting?
For me, this only appears to happen when copying from a plaintext document to a rich text document (or vice versa.)

Make sure your source and destination text programs are the same format. If your setup is like mine, Evernote is rich text, so make sure you haven't set TextEdit to plaintext. With TextEdit open, go to 'Format > Make Rich Text' (shortcut: Shift + Command + T).

If your setup is different then mine, then maybe Evernote is plaintext. Which would mean, if this error is still occurring, that your TextEdit is set to rich text. Use the same menu shortcut to switch it (or go to 'Format > Make Rich Text'.)

In either case, switching your TextEdit document will result in the table zigzagging like you mentioned. But if, going forward, you format your tables in the text mode that matches your destination's text mode, the problem should abate.