> Are countries within the EU sovereign, or is the EU sovereign?

They’re both sovereign. When a king signs a treaty binding them in some way, they give up some sovereignty. But they don’t usually cease being sovereign.

The EU can punish or withhold benefits at will from misbehaving states. Sovereignty isn't the issue when you are locked onto the teat with superglue; if the cow witholds that precious milk you will starve, sovereign or not.