start as a flooded engine for 97 corolla, why and how to fix

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My 97 Toyota Corolla (136K) can start only if following the instruction, described in the manual, for a flooded engine: turn the key to ignition and push the gas pedal the same time for ~ 5 seconds, release both, then start the car. Sometimes, have to repeat several times. It seems the problem started this summer, but was minor. After I changed engine oil(5W30, new england area) and filter last weekend (I have been doing this for the past 10 years), the problem came out on a wet Monday morning. I thought the battery died out, then got a new one. The problem is the same. I am the original owner, and the car has never been in water. The scheduled maintenances were done before 50K. Except for regular oil exchange, no any other service was done after 50K. I love this car, but just afraid to take the car to any garage because I have been ripped at least twice by the dears (they once charged me ~$250 for a spark plug replacement on my other car!). Can somebody explain to me why this happens? How to fix it? Many thanks in advance!