Layover and change airline in Shanghai

Your itinerary is definitely eligible for 24 hour visa free transit and you don't need any kind of visa. You will print out your flight tickets and have those printouts with you. Also, you will declare at check-in in Jeju that you will travel to Dubai via Shanghai using 24 hour visa free transit. Your tickets will be inspected and after a few minutes, you will be allowed to board the plane.

Upon arrival in Shanghai, under the assumption that you will travel on separate tickets, you will approach 24 hour visa free transit counter where you will apply for a temporary entry permit to be allowed to exit the airport because you will have to collect your luggage and also to perform a recheck. This is all fine, but you will have to be very agile and fast because Shanghai Pudong airport could be quite time consuming.