Start with SoM's suggestions - check everything, inside and out.
The fact that you have some play in the height adjusting screws for the bridge is normal, and with strings in place and tuned there should be more than enough pressure to keep these from moving. It's extremely unlikely that these two screws are causing your problem. (The recent discussion about bridge play was for individual saddles, riding on the intonation screws.)
As for the fit of your E string - be careful. I would recommend using a needle file (or better yet a nut slot file, but not many of us have those). Ideally, you'd like a half-circle groove that precisely fits the string. However, you don't want to cut it too deep because then the action won't be consistent with the other strings. You also want to have a well-defined witness point, i.e. you do not want to have a rounded, curved entry on the neck side of the saddle, and the back side of the slot has to be deep enough so that the string is fully seated on the front (neck) side.
If you're patient, have good eyes (or a magnifier), and understand the concepts, it's not too hard - otherwise, have someone do it for you.
-Bob