Turning on AC either resets thermostat or blows fuse
First I would make sure all the wires are hooked up correctly. Look at the thermostat, what color is hooked to R, Y, C, G,W and whatever else you have. Then go to the furnace and make sure those same wires are hooked up the same at the furnace. There is a cable going to the condensing unit outside from the furnace. One wire should be hooked to Y and the other to C on the furnace. The other end should be hooked to the contractor outside.  Also make sure none of these wires are grounding out. Make sure the new contractor is 24 volts at the coil. They make them with 120 and 240 volt coils. You say the contractor is white? The only white ones I have seen are special purpose. Usually they are brown or black. I would be interested in seeing a picture with the label. My guess is something got wired back up wrong with all the fiddling.