Kind of depends on what you want/need. keeping it in the x57 family will be less work and likely feed better.
458x 57 a brush thumper
11x 57 a 44 cal-ish - something like a 444 in a bolt gun
10x57 never heard of this one but hey
**The above might not feed well out of the mag
9.3x57 a...
If you think its from powder residue, I'd try scalding hot water with a good bit of Dawn dish soap. If congealed oil, parts/brake cleaner. Either way a good soak will likely be needed.
As to the ewe harvest, Nevada has had some ewe hunts in units that have known persistent disease issues (pneumonia). They want to keep densities lower to prevent transmission/outbreaks. Since these sheep are known to be diseased, they don't want to move them and cause issues elsewhere. If the...
The Savage 24 is the most common and economical, Tikka, Heym, an others made combination/cape guns as well. Or you could go with a drilling- double shotgun over a rifle barrel.
Seems to be the norm down in NM, even some of the larger county roads were gated and locked. It was really hard to figure out what was public and what was not.
Henry Martini or a small frame rolling block? Contender carbine. All will end up being a bit of money- think id just go with a 10-22 with a basic scope.
Here is what I would do- proceed with your own caution/comfort level.
Take standard 30-06 brass (once fired is fine- and not from this gun) and run a .338 and evenn maybe a .358 expander ball through the neck. Slowly neck size the brass with a set of 30-06 (AI?) dies and stop then the bolt...
Its always nice to print a copy of the airlines rules, and highlight the pertinent parts. I've had some try to demand I do stuff that is wrong, having the rules handy shuts then right down.