Drying:

A shop vac comes to mind for the top of the carpet. Then you need to remove the rear seat cushion, remove the plastic trim at the bottom of the door (carefully), lift the carpet, and Then:

lift the rubber sound barrier. Note the insulation under the rubber, it absorbs more than its weight in water. Remove as much free water as you can. I used a Sunday newspaper without the glossy ads and placed it between the insulation and the metal floor. Then pushed the rubber side down to compress the insulation and let the newspaper mop it up. It may take more than one newspaper! Check with your neighbors before they 'recycle' it.

I agree with HF63, a vent is probably clogged in front of the windshield.

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My back feels better when I sit in a Volvo seat