George, this is where an irrigation background is really helpful. I often find exactly what you're talking about in irrigation control wires where the solenoid ohms out fine and if you disconnect it you get proper voltage but as soon as you connect the solenoid up for a real load, the voltage drops to nothing. It's always a bad splice or a wire break inside the jacket. I can usually find the problem pretty quickly but sometimes it means running a new wire.
To the original question, I think you're seeing corrosion because of water intrusion (as others have said). You can probably get the wire clean enough to solder with a mild acid, like CLR, but if you're seeing so much corrosion that you need to acid wash the wire, I wouldn't use it. Soldering is fine if you want to take the time to do it but making the connection water tight is more important. No matter how you make your connection, I'd suggest you start using the gasket lined heat shrink to protect them. I'm amazed at how much better the wires I've used the heat shrink on look than those where I used other methods (like grease filled wire nuts) after the same amount of time in the ground.