A couple of days ago I was at the range and a fellow a few stations over was shooting a 300WM he'd gotten passed down from his dad and tried shooting a leftover box of original (about 40 year old) ammo. First few rounds, he said, seemed to go ok. Next one blew some material on his face and the recoil was violent enough to bang the scope against his eye, draw blood, and jam the bolt to where it could not be opened by applying significant hand pressure. Overall, the ammo showed some tarnish and an overall brown discoloring, or fine dusty residue, on the outside of the cartridges (significantly worse than what you show in your picture). After the incident, closer inspection of the previously fired and ejected cartridges showed one with a split neck. However, even more alarming, several of the remaining unfired cartridges also had noticeably visible cracks on the necks. Overall, he was a lucky guy... hope his rifle is also okay!
Jowens, you are probably okay to shoot what you have as your ammo is in much better visible shape than this guy's. However, please stay safe and look each round over carefully before shooting to confirm that there is no unexpected stuff going on. Stay safe!