What are you using for ammunition? I had the exact same problem with my pre rail 3913TSW and my 3953. I was using 115gr rounds at the time. I found that moving to 124gr improved both issues greatly (I still pull a bit to the left, but that's me). It's even less of an issue with 147gr.
Here is an picture of my target with my new pre rail 3953TSW.
As you can see, I still do pull a bit to my right, but not as much as I used to.
This is an earlier target with my pre rail 3913TSW with 115gr rounds.
I can't comment on the "heavy" feeling of the trigger, all of my 3913s have a good trigger pull. The DAO are a bit heavier, but once you get used to that, it's not bad. Actually similar to a DA trigger on a K frame. Which I think was the goal there.
Like you, I was incredibly frustrated until I has an "AH HAH" moment with regard to the ammunition.