Ftm:labor question

Sorry, i know this isn't probably the correct group to ask this... But I'm a ftm and I'm wondering if any of you ladies have gone into labor without having your belly "drop". I'm 40 wks and 4 days I'm slowly losing my mucus plug, have had no signs of pain/ contractions and my belly is still pretty high. My baby has been head down and engaged.
Can i go to the hospital to be induced or do i need to wait until I get pain and/or my water breaks?
Thanks in advance.