Do you know how the water is getting onto the carpet? When I had this problem the water was dripping onto the carpet from just behind the glove box (directly beneath the windshield seal). The reason I ask is that if the water isn't seeping into the carpet from somewhere further forward (firewall or heater core for example) you could at least put down an all weather mat to stop the soaking effect.

The only-when-you're-sitting symptom likely just means that you have a very slow leak - water has to pool and sit on top of the loose seam for awhile before gravity takes effect. If you're in motion the water doesn't have long enough to go through.