The reason this happened is because the shoulder on the .280 Rem is slightly farther forward than the other members of the .30-06 family. They did this to eliminate the possibility that someone would chamber a .280 Rem in a .270 Win and proceed to blow themself up. With the shoulder 0.05" farther forward the 280 Rem can't chamber in a 270 chamber.
The problem you're having is that the shoulder on the .30-06 brass is too far back, so when you size it in a .280 Rem die, the die only resized the case back to where the shoulder is supposed to start on the 280, leaving a hump or ringshaped bulge. To finish making .280 Rem cases from those .30-06 cases, you'll need to fireform them. To do this you can either load up some lower powered rounds, preferably with one of the faster burning powders suggested in your load manual, and take them to the range and fire them. When you do extract them they will be perfectly formed .280 Rem cases that match your chambers dimensions perfectly. You can also use the "creme of wheat" method, which forms the cases without using a bullet. I think for your purposes being that you only have to fireform them just a little bit and you're not shooting an ultra-high velocity round that burns barrels fast, I'd just load them with bullets and 7-10% reduced starting loads and let them rip. You'll actually probably see pretty decent accuracy.
I did this when reforming 7mm Rem Mag cases to .257 Wby. You run them through the .257 die and you get a 99% formed case, but the outer shoulder is still not completely radiused. I just loaded them with starting loads and shot them, now I have perfectly formed 257 Wby cases, and my accuracy was only 1/2" bigger than the actual Wby cases I also was using (and about 1/3 the cost

). The necks are a little short compared to the very long Wby necks, but they shoot great.
Mike