mosschops ikwym about the show - I had one for about 2 weeks prior to labour with both of mine!! It is a good sign though as I am sure you know - your cervix is softening. Pupuce is right - there is a point to waiting - it may well have an effect on your labour.
especially if by your dates you will only be 1 week over. Wonder why the hospital are going by "their" dates and not yours? Presumably you booked birthing centre hoping for a fairly natural birth with less intervention - your chances of this do slide with an induction. Your chances will remain high though if you follow your babys cues and birth when you are both ready. You don't "have to take" any induction date. It is your labour and your body and you have the right to go into labour spontaneously!
I know it is all very well for me to sit here and egg you on to wait - I do remember what it feels like to be 9 months + (both mine were 14 days over) but it helped me to visualise my baby and think about it from the babys point of view - then weigh up the pros and cons.
I hope none of this sounds harsh or pressurising, I don't mean it to be that way - just the opposite. Its just that I have met so many women who have had births that have not turned out the way they planned after inductions, and memories of birth stay with you all your life. Of course your health and the baby's health are more important than your experience in labour and birth, but a good birth means a quick recovery in most cases.
Really really hoping you are in labour NOW or will be very soon. Hope this doesn't seem like a rant.
Love and support x