Can I remove my old core support and replace it without draining the radiator? Can someone give me a quick run through on the steps of removing the core support?

You have to take off everything on the front of the car in front of the radiator support. You can leave the fenders. Its a good idea to turn a rim on its side and have some 2x4s to support the radiator, or cinderblocks, something to the likings of that, because if the car is an automatic you are going to have metal tranny lines running into the radiator which you dont want to bend, at the age of our cars they break easily. I also recommend getting new bushings for the front end when you do this. Just unbolt the old core support and bolt the other one right back in.