Having been a recent switcher to Android and the Galaxy S3 I thought at first that it was me doing something wrong. No matter what settings I tried my S3 would always answer my calls after a few seconds, I went through all the usual settings and even tried headset on/off and eventually even had more experienced users helping me diagnose the problem.
Well, I'm happy to report that I finally nailed what was causing the issues for me, it was believe it or not the fact that I had my voice mail going to Google Voice. As soon as I turned that off and went back to letting AT&T handle my voice mail the phone immediately let me decide if I wanted to answer the call or not.
So if you have tried and tried and tried to get it to behave, I ask you this question ... are you using Google Voice? If so, try turning it off and see if the phone waits for you rather than puts you in awkward situations where you are now talking to people without realizing it :)
