To induce or not too, due date confusion

Emily

Im 39 maybe 40 weeks pregnant. My doctor has been going by the date of my last period that I originally gave before I was sure, therefore using April 2nd as my due date. Looking back through my period tracking app I discovered it was 5 days before the date I listed. My first ultrasound puts my due date at March 28th. That's 5 days difference and he would like to induce at 40 weeks, which would have been today if he went by the ultrasound or tomorrow if by the CORRECT last period date (June 22). On top of that, my grandfather is extremely sick, he doesn't have much longer and my parents told me his nurses think he's waiting to go after I have my baby. Which makes me feel even more pressure to induce if my little man doesn't come naturally by this weekend. My last period was only 3 days long so I'm also paranoid that it was bleeding from implantation and maybe my last period was actually in May- which puts me wayyy overdue. I'm not sure what to do at this point and my husband thinks I'm being impatient and paranoid. Any thoughts or suggestions from veteran moms?