RE: 260 vs 280, Pros/Cons
First of all, let me reassure you that neither the 6.5x55 nor the 7mm-08 are anywhere near obsolete. They are two of the finest cartridges available for deer size game.
Now on to your question. I'd would say that for dual-purpose usage such as you imply, the .260 Rem would be my pick hands down. It is light-recoiling so you can get lots of practice, and it will take whitetails cleanly to as far out as you would want to shoot one. There is a good selection of 6.5mm bullets available if you do take up handloading, and it is capable of wonderful accuracy.
The .280 Rem is also great, but while it is an excellent whitetail cartridge, it may be a bit much for coyotes. You will also notice more recoil with this round.
Good Dogwork and Good Hunting