Sorry but I am unable to replicate. When I pull my headphones the song pauses automatically. When I reconnect the headphones I have to click on the play icon for the song to start playing and it picks up from where I left it.
You are not going insane but something is wrong with your phone. I would start with a forced restart.
Hold down the Home and Wake/Sleep buttons at the same time for about 15-20 seconds until the Apple logo appears. You will not lose anything.
If not I would try resetting all settings (but be aware that you would have to reconfigure all). The alternative is a system restore, first as new to test and then from backup.