Does a capacitor only releases energy once it's fully charged up?

No, or at least not in the normal way of using one.

A capacitor will release energy if it has any charge in it (not necessarily full charge). To release energy, a capacitor needs a suitable current.

Does a capacitor only releases energy once it's fully charged up?

 

A capacitor releases energy any time the voltage across whatever it is connected to is less than the voltage across the terminals.

Consider the following:

Charge a capacitor to 1V by attaching it to a power source momentarily.

If you attach it across a piece of wire, or LED, it will release energy into the circuit.

If you attach the positive terminal to the positive terminal, (and - to -) of a 1.4V batter, the capacitor will instead continue charging.

 

Capacitors that are charged will also leak a small amount of energy because the dielectric is not a perfect insulator. The amount lost depends on how charged the capacitor is. This is (one of the reasons) why we don't use capacitors instead of batteries for long term storage.