I think I had the same problem once, adjusting the packing nut made it worse, but give it a try.
the picture shown is a union, you do not really need that..
You need a coupler called repair coupler, it's the same as a coupler but without the dimple/stop so it can slide anywhere on the pipe.
Cut valve, but you can actually unsolder the valve from the pipe and clean the pipe and reuse without losing any length, but that's up to you. Because pipes are so expensive now, if you choose to unsolder, you only need the new valve and a coupler from home depot. It should be under $10.
Solder the new valve on, then slide the repair coupler to the middle of the disjoint and solder it.
watch this video