Basic Mauser 98....
And you can't find a smith in the Seattle area? Either things have changed alot, or you aren't looking in the right places. Maybe both.
I can't help you in the Seattle area, but Don Manning at Shooter Supply in Yakima could take care of it. So could I and a couple of guys I know who are not 'smiths for a living.
I can't really tell from the pictures, but the extractor "claw" should reach nearly to the bottom of the groove in the case, with the other side of the case tight up against the projection of the bolt face. The round should be held snugly, and should not fall off the bolt face if lifted.
Mauser rifles are intended to be loaded from the magazine. Rounds feed up from the magazine, under the claw of the extractor, on their way into the chamber.
Putting a round in the chamber and then closing the bolt on it forces the extractor to flex in order to snap over the case rim and into the extraction groove. Some Mausers have a clearance cut, and the tolerances to permit this. Some do not.
The Mauser extractor is spring steel, and can be bent/lose tension or even be broken as a result of being forced over a case rim, if the rifle doesn't have the clearance to allow it.
You said you tried a different extractor, so what you might have is a bolt face that has been "opened up" to hold a larger diameter case, something done for "magnum" conversions. Or the bolt face could just be very worn.
You definitely need to find a smith to do a headspace check and verify the chambering BEFORE you try to shoot it. I know there are people who can do this in Yakima, TriCities, and Spokane, and there used to be people like that in the Seattle area. I'm pretty sure Al Biesen has retired (and likely passed on), but he was a nationally recognised custom rifle builder, and he lived over in that area.
Go to the Washington Arms Collector show, they have one every month in Puyallup. Somebody there will certainly be able to help you find a smith to get your rifle checked by someone who knows what he is doing. I'll ask some folks I know if they know anybody in your area, but I won't be able to see them before next week.
Please don't shoot the rifle until you know for certain why it isn't working correctly. Be safe, not sorry. Good Luck.