Have an Apple iPod Classic that uses the old-style 30 pin connector.
Trouble is, the Sony Dream Machine docking station I bought has an 8 pin Lightning port.
An adapter I bought on eBay fits, but doesn't work.
Can't seem to find anything on Amazon.
Can anybody recommend an adapter that will allow both charging and music playing?
No such adapter exists officially. The official Lighting-to-30pin-dock adapters are to allow you to use newer devices on older docks, not vice versa. The crap on eBay is just a "dumb" adapter that passes through the power and USB. The ebay listings say "Supports charging (with a wall adapter) and data sync (with a computer) but will not be recognized by any other device" so I'm not sure why you expected it to work.
Since Lightning is digital only (unlike the old Dock connector that was digital and analog), and it's likely using a proprietary Apple protocol to receive said digital audio, there's no practical way to feed in audio to it.
Go back on eBay, but instead, buy a speaker dock with the 30-pin Dock connector. Since they're obsolete now, and millions were sold, they should be pretty cheap. Similarly, you could find one at a thrift store.