Plumbing – Water shut off to solder on an adapter

I think what is leaking is the drain. Does it look like the object sticking out to the side in the below picture? If so, this is used to drain water from the house side of the supply when the valve is off.

It sounds like you over tightened it in the first place which started the leaking. If you've broken only the cap, you might be able to find a replacement. If the stem and threads are damaged, long-term you will need to replace the entire valve. As a short-term fix, you just need to block it some how. Marine epoxy is handy for emergency plumbing repairs.

This answer may be inaccurate because I assume you have the most common type of shutoff valve, which is a seat and washer type (v. gate valve or ball valve) and an oval handle (called, I think, a wrench pad).

First, shut off the main valve, so if you do break a pipe you won't flood.

Try turning the handle with a wrench while maintaining a counter-rotating force on the valve body with another wrench. This usually works for me, although the valve remains stiff and sometimes I need two wrenches to close it again.

If that doesn't work, try removing the compression nut. Here also you want two wrenches, a proper size crescent wrench for the nut and a pipe wrench for the valve body. As the nut starts to move, turn the handle with it and remove the handle, stem, and nut together. Once you have them out you can separate them without stressing your pipes. You will probably find that the packing has stuck to the stem.

Of course if you do break the valve or any pipes then you will have to replace them but at least you tried the easy methods first.