1997 Dodge Intrepid Oil Pump

1997 Dodge Intrepid 6 cyl Front Wheel Drive Automatic

I owned a '97 Dodge Intrepid for awhile. It ran fine but after awhile it felt like the engine was shaking and then finally one day I eased on to the gas and something broke under the hood. Now everytime the car runs there is a non-stop clicking noise. Everyone says it sounds like the oil pump went out. The car still works but there is still extreme shaking and that non-stop click. Anyway, I got rid of that car. It is almost winter now and I JUST have purchased a '97 dodge intrepid sport and whenever I start the car I hear that same click noise that the first one but it goes away within 45 seconds and it even shakes badly too. The wierd thing is that once the car warms up the shaking slows down and eventually stops after awhile. So should I be worried that the same thing will happen to this one? Or is it just something that will go away eventually? Because no one has told me 100% what that clicking problem is. And I don't know if the cold weather has anything to do with it. Maybe the cold is thickening the oil so its hard for the oil pump to deliver oil. But should I be worried about that?