I am guessing this falls under the topic of Homework, but since you are trying to make something I am tempted to side-step that restriction here.
Your basic "thing" is pretty close. Part of the reason it jerks is that there is no means of maintaining the sliding plate parallel with the base plate during the travel. Also the force to overcome the beginning movement of the syringe plungers is probably higher than the force required to maintain a steady flow out of the syringes once moving.
So, first things first, in the final project, how will you be pushing the top plate down? Fingers, air cylinder, electric motor, 4 large bricks, car jack?
We can get to the other issues in a while.
Generally, I will not give you the answer to your question, but I *will* guide you into discovering how to solve this yourself.