Who is responsible for the water damage caused by a faulty part in a new dishwasher?

The manufacturer and the store where you purchased the dishwasher are both liable for the damage caused by the defective product.  Your lawsuit against the manufacturer and store would have two causes of action (negligence and strict liability).

Negligence is the failure to exercise due care (that degree of care that a reasonable manufacturer would have exercised under the same or similar circumstances to produce a product that is not defective).  The seller (store) is liable even if it could not have known that the product was defective.

Strict liability is liability whether or not due care was exercised.

Your damages (the amount of compensation you are seeking in your lawsuit) would be the cost of repairs to your house for the water damage.  You will need to mitigate (minimize) damages by selecting someone to repair the damage whose charges are comparable to others in the area providing this repair service.  If you were to select the most expensive repair service you could find to repair the water damage, your damages would be reduced accordingly.

You would file one lawsuit naming both the manufacturer and the store as defendants.  Your lawsuit would have two causes of action (claims); one for negligence and the other for strict liability.