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.
Almost all thermostats have a FAN switch, that can be set to ON or AUTO. In the AUTO position, the blower comes on when the furnace needs it (during heat/cool calls). In the ON position, the blower will always be on. This is usually controlled by the G terminal on the furnace and thermostat.
I'm not sure exactly what you're trying to do, so I'll cover a couple different setups
Blower on when you want
For this situation, simply connect the G terminal on the thermostat to the G terminal in the furnace. When you want the blower on, set the thermostats FAN setting to ON.
Blower on when house is too warm
If you want the blower to come on when the temperature in the house raises above a set temperature, you'll simply connect the Y terminal of the thermostat to the G terminal in the furnace. This way whenever the thermostat calls for cool, the blower fan will come on.
Depending on the furnace, you may be able to connect the Y terminal on the thermostat to the Y terminal in the furnace, and simply not connect any control wires from the furnace to the A/C unit. Most furnaces activate the blower after a predetermined (possibly programmable) delay, when the thermostat calls for cool.
Blower speed (Multi-speed blower)
Blower speed may vary based on the different functions. For example, heat might be medium low, fan might be low, while cool may be high speed. So you'll want to take that into consideration as well.
Without more information about your specific equipment, and what exactly you're trying to accomplish. It's difficult to provide more accurate, specific information.
Best Answer
These are called "saddle valves" or "vampire taps". They just poke a hole in the pipe with a little pressure around the hole to seal it. They almost always leak after some time but they are quick and easy to install.
The right way is to cut the pipe and install a tee fitting that then supplies a standard valve. You can use push-to-fit connectors these days to make the job simple or you can used a traditional soldering approach.
Because of the number of fittings in that small space and the fact that the pipe is damaged with two saddle valves, you'll be replacing that entire bendy section.