Try a odd complete graph like Your Hamiltonian cycle can just go around the outside, but the Euler cycle must obviously hit all edges. A simple loop is not going to work. You're going to have to have more edges than in your Hamiltonian cycle.

Euler cycles include all edges. Vertices come along for the ride, perhaps, but that's not in the definition. Similarly, Hamiltonian cycles include all vertices. It may get all the edges as well, but that's not in the definition.