I can understand how you feel. I am also pregnant with my third and am suffering severe back, hip, and pelvic pain with little to no sleep.
The problem with trying to naturally induce is that the methods only work if your body is already starting labour. It is the baby's lungs that release a chemical signaling to the woman's body that the baby is ready to come. So you will just wear yourself out even more trying things to bring on labour. Not to mention the stress and frustration you will feel every time something doesn't work.
As for the ER doctors inducing labour, they can't do that without an okay from your doctor, unless it was an extreme emergency and then they would just wheel you in for a c-section.
Your baby is healthy and a great size, for the age it is. I know you are miserable but just hang in there. This will all become a distant memory once your third one comes along. I know how hard it is to be experiencing these symptoms while trying to raise two other kids.
You have come this far, just hang in there, for the baby's sake. Your baby would still be considered premature at this time and would still spend time in the NICU.
Like I said the ER docs can't do anything without the okay from your doctor first, so going to the ER will really do nothing but stress you out more.
Good luck!