Soontobemama you don't have to go down the sweep and/or induction route, at least not yet! I feel your pain, I went to 40+5 with my first and 40+11 last time which meant I had at least three weeks of "still here then?" and "not long to go" and the constant texts and calls to ask if there was any signs of labour by mil which sent me demented.
The hospital started to nag about induction from roughly 41 weeks. I refused, especially as the dating scan at 12 weeks had put my edd forward by a week (I knew when I'd conceived as we were using ovulation sticks) and by 40+10 they were getting really arsey with me for not meekly complying. I did agree to a sweep which was painful and I don't really think that it did anything.
Went into labour 40+11 naturally five minutes after dtd with my partner, no interventions needed. Lots of walking and nipple stimulation strengthened and established the contractions. No amount of curry or pineapple helped either time. She was very petite, 7lb, still had lanugo all over her. Definitely not an overdue baby.
In some places in the world you aren't even considered overdue til 42 weeks and their intervention rates are much lower than us.
This time around I've told everyone I will go overdue. I'm meant to be having a homebirth which adds a little pressure as I know they'll be less inclined for me to have one if I get too overdue, and I had to bloody fight for it as it is.
Hang in there, they can do expectant management to check daily/every other day how you, baby and placenta are doing. It's much easier for them to just induce you, so they probably won't offer to do this for you. But if you have no other health problems and good fetal movement then go for it, sod their arbitrary rules and estimated dates.