EDIT: I tried using Deepbit instead of slush's pool, and all of the invalid/stale results are gone. Is this something to do with using slush's pool, or does the deepbit server work differently? They say that "no block confirmation is needed", but does that mean that I'm still submitting invalid blocks, and that they aren't checking them?
You aren't submitting blocks to pool, you are submitting shares (hashes for difficulty 1). Block confirmation is not related to shares.
BTW, what operating system are you using ? If it's Windows, then you need older version of python miner to work with 5970s.