After all the celebrating over the Shiny New Job, it looks like I'm not going to get it after all. They only offered me the job "subject to references", and my reference from my current employer told them how much sick leave I'd had recently.
I explained the situation, and while they did say that they appreciated me being honest with them (more fool me, yes, but they phoned me out of the blue to ask about this and I didn't have time to think of a good lie, and I did think honesty would be the best policy) they pointed out that plenty of other people are stuck in jobs they hate and they don't take sick leave.
They're waiting until they get my second reference before they make a decision (presumably because if they offer a job "subject to references" they're legally obliged to wait until they've seen both references until they turn me down). So at this rate I'll probably get the final "no thanks" decision on the day of my birthday party.
(Please, no one-word "hugs" responses or equivalent. I know it's well-meant but it's really not helping.)