From the original post: "Other valves I've dealt with have a nut right behind the handle but these don't." I agree with others that these valves should be replaced at this time. You will avoid further troubles later.
Looks like the copper has male threaded connection. Why mess with cutting, soldering, etc... Just turn off main unscrew valve and replace with new one for $7 each. I would go to braided feed lines also. $25 and 15 minutes an you are done and upgraded.