Coolant wont stay in overflow

I have a 2001 Chrysler Sebring LX Sedan with a 2.4 DOHC ("X" VIN) 4 cylinder engine. The car has 128,000 miles on it with no major repairs made to it. The car began overheating 2 weeks ago so I replaced the thermostat. That took care of the overheating but then I noticed air coming into the overflow. Since there wasn't any noticable water in the oil, I decided to replace the radiator pressure cap. Everything is working great. The car is not overheating and no bubbling in the overflow. The problem is, I can't keep water in the overflow. I've been under the car and there aren't any visible signs of where the water is going. All of the hoses and main drive belts are dry. I've contemplated changing the water pump but am hesitant to do that as part of a "trial and error" repair because of the cost. Wouldn't a bad water pump also cause the car to continue to overheat? I've tried running the car with the overflow empty and still can't get it to run hot even with 90+ degree days and the air conditioner on high. I'm stumped. Any help would be appreciated.