It is unlikely you will go early so sorry to burst your bubble.

What you are feeling is perfectly normal in my opinion. I was so over both pregnancies by 36 weeks. Both mine went at least 9 days over EDD.

Being tired is probably just a symptom of being in late pregnancy. Nothing to do with impending labour. I did not drop with either pregnancy until I was in labour and never had any braxton hicks. The first labour was induced as DS1 was showing no interest in leaving his comfy spot. The first indication of my labour with DS2 was a mild contractions the day before I gave birth which were regular but reasonably faint. I was able to sleep fairly well through the night, waking up a couple of times with the contractions but not enough to be concerned and next morning at breakfast had the first major one.

Have you started nesting? I have been told (though it did not happen to me) that when you start nesting, labour is not far off.