Hi Sandy, That certainly does sound like a hardware issue. If you signed up for the warranty plan through MTS you can bring it into a connect store and they can help you make a claim for the device. Otherwise I would suggest bringing it into Apple to have them take a look at it, see if it can be repaired.
Did you upgrade your iPhone 5C to the latest iOS 9? The newer platform will actually respond to 'almost' touches where your finger is above the screen close enough for the phone to sense it. If the screen is on then this could be what is happening with your device.