As pointed out in the comments the rigid rotating molecule only gains kinetic energy when it rotates not potential energy.
The degeneracy describes the fact that some levels have exactly the same energy and this depends the value of the angular momentum rotational quantum number $J$. The number of degenerate levels is given by the multiplicity $2J+1$. The $J$ quantum number has values $J=0,1, 2,3,\cdots$. A second quantum number $m_j=-J\cdots J $ describes each of the degenerate levels and 'orientation in space'. Thus if $J=2$ then $m_j$ has five values $m_j=-2,-1,0, 1,2$. Normally the $m_j$ levels cannot be distinguished by experiment because they are degenerate in energy, but in a magnetic or electric field these energy levels can be made to split (degeneracy is removed but multiplicity remains the same) and transitions between them measured. These effects are called the Zeeman and Stark effect respectively. (Historically the Zeeman effect was of great importance and showed that electrons have spin angular momentum, unknown hitherto.)
As the molecule is a quantum object some care is needed in interpreting what is meant by 'rotation'. When $J=0$ the molecule has no kinetic energy, i.e. is stationary and so has a wavefunction like that of an s orbital. To satisfy the Heisenberg Uncertainty Principle the molecule has an equal probability of pointing in any direction in space, i.e. the wavefunction looks like a sphere. When $J >0$ the molecule has some rotational kinetic energy but it now becomes more difficult to imagine but when $J=1$ the molecule can 'rotate' in any of three directions, with wavefunctions shaped like p orbitals. When $J=2$ the wavefunctions look like those of d orbitals and so on.
(Quantum mechanics often presents problems because we are so familiar in thinking of how every day objects move, e.g. a rotating wheel, and its easy and sometimes useful to imagine molecules do the same thing, but this is misleading as the wavefunction describes what happens and this often does not seem to make physical sense. )