The "double end" accommodates a variety of attachments. The idea is that you cut off the end that doesn't match your setup. Then the cable looks like a brake cable :-)
So, what you should do it to remove the old cable and match the end to the new cable. Then cut off the end that you don't need (cut near the "wrong" end so that you preserve the cable length). Now you'll be able to thread the cable through the housing (sheath) and make your repair.
When you remove the old cable pay attention to how the ends are attached. You'll put it back the same way.
If you don't have / can't afford the investment in a "proper" shearing type cable cutter mentioned in the comment, you will improve your chances is you cut at the very end of the cable up against the end you are removing. The metal that formed the end may help to support the cable strands and create a cleaner cut. If the end does get mangled, try to reform the end of the cable by twisting the strands back into place. Here's an example of the kind of cable cutter you're looking for (in this case a Park Tool CN-10).