First of all, Rathnait, you're not alone. I loathed and despised As Good As It Gets and I booed and jeered when Helen Hunt won the Oscar, which should have gone to Helena Bonham Carter for Wings Of The Dove.
But as far as tearjerkers go, I realise I am a little late with this, but Dancer In The Dark. I have never cried so hard at a movie before or (mercifully) since. I was still crying 30 minutes later in the cab on the way home. It destroyed me.
Nothing precious, plain to see, don't make a fuss over me. Not loud, not soft, but somewhere inbetween. Say sorry, just let it be the word you mean.