When you want to dig into problems the first thing you should do is select all of the layers you are having trouble with, press the U key twice to reveal all modified properties and analyze at what you see in the timeline. I'm just guessing here because you have given us no indication of your workflow but it makes sense that you have applied the layer style Drop Shadow to both layers. I'm just guessing that you used Layer Styles and not the Effect because of the title of your post.
Since you have already done the animation and now want to add a drop shadow the easiest thing to do would be to pre-compose the two shape layers and then apply a layer style or an effect to the pre-comp. It looks like you have Drop Shadow applied to all of the text layers also so it would make sense to pre-compose all of the layers you want to have
I probably would have used a rounded rectangle, converted the rounded rectangle to a Bezier Path and then animated the shape of the path and used trim paths to get the circle to the rounded rectangle animation on a single shape layer but I would definitely have pre-composed before adding the same effect to all layers.
Ok thank you!