So I did a water test on the hydrometer, and indeed it was out by 4 points (1.004). So I took another test of the oktoberfest with my second (working thermometer) and it measured 1.015. So I've put it in the fridge to lager. I gave the sample a taste and any oily-slick texture is no longer there.
My only concern now (and it's probably rather trite) is that it seems a bit darker than I expected.
I'm glad I found out now that the first hydrometer I had was out of whack. It's probably given me a few misreadings on a few batches. I guess I'll have to pick up another one for backup, although I have yet to physically break one... knock on wood.