You could aways come down with something for those lase 12 days and use up that sick time. I don't condone the idea but I have known plenty of people that take all their vacation days in the last 3 or 4 weeks or so then they work 2 days at a time so they don't need a doctors note, yes, where I work if you are out for more than 2 days you don't get paid without a doctors note, to limit the amount of time they need to be there.
I know it doesn't put cash in your pocket but it at least gives you time to yourself to prepare. Changing jobs can be as life altering as getting married/having kids/etc.
And congrats on the new job.