Mag lips will need tweaking, but oddly enough the two symptoms you report require tweaking in opposite directions. To keep the first (and subsequent) rounds from popping out you will need to bend the lip inward a little to hold the round in place. Once you get that consistent, you will either have to play with just the front part of the lip to fix the second issue of the bullet catching on the face of the breech. Two options here:

1. You can try to tweak just the front part of the lip slightly wider to allow the nose of the round to sit higher in the magazine.
2. You can file on the front portion of the lip to open it up and allow the nose of the round to sit higher.

Savage never offered the .280 or 280AI so naturally they don't have a specific magazine for it and you are going to have to make some modifications to get it to work.