You can get a little copper thing from the store it that sells fishing gear, and crimp that on. Auchencrow explained this and even sent me a few. The best crimper tool is a Bell Systems tool you can often find on eBay.
They're usually brass, not copper. IIRC it's the
#3 leader sleeve you want. The crimping tool looks like this:
The picture makes it look larger and more imposing than it actually is (about 6" overall length), but once it crimps something, it stays crimped.
Alternately you can get an anchor that bolts a new barrel adjuster to the cable with an eye bolt. Your bike shop should have those. They may prefer to sell you a whole cable kit, which will include one.
Yes, replacement shift cables are readily available and work fine if you're not stuck on the OEM look.