First of all, your shoulder blades should be back and down during the set. Your upper back should be arched. This pushes your chest up. Do not lose your arch. Once your chest caves, you won't be able to fully activate it and other muscles start taking over.
Second, your elbows should start flaring once you start pushing the weight up. This is how you use your chest. If you are only lowering to 2/3 of the range of motion, i.e, around 2 inches above your chest, you don't have to tuck your elbows so much.
Third, when you lower the dumbbells during warm ups, pause for a 1 count when you feel the stretch in your chest at the bottom. Use your chest to explode the weight up by contracting hard. Just do this during your warm ups to get into the groove and do normal reps for your working sets.