The Overall Movement
The angle \(\theta\) has two meanings in our inertial frame. It is both coordinate and angle of the molecular axis to the electric field. Using the expression for the conserved angular momentum \(p_{\varphi}\), we find a relation between the angular velocity \(\dot{\varphi}\) and the angle \(\theta\):\[\begin{eqnarray*}\dot{\varphi} & = & \frac{p_{\varphi}}{mR^{2}\sin^{2}\theta}\end{eqnarray*}\] Note that neither \(\theta\) nor \(\dot{\varphi}\) is constant in general!
So, how can we employ this relationship to understand the movement? We may replace \(\dot{\varphi}\) entirely with terms in \(\theta\). Next we can use the second conserved quantity, namely the energy \(\mathrm{E}\). Remembering that \(\mathrm{E}=T+V\) we find \[\begin{eqnarray*}\mathrm{E} & = & \frac{mR^{2}}{2}\left(\dot{\theta}^{2}+\sin^{2}\theta\dot{\varphi}^{2}\right)-qRE_{0}\cos\theta\\& = & \frac{mR^{2}}{2}\left(\dot{\theta}^{2}+\frac{p_{\varphi}^{2}}{m^{2}R^{4}\sin^{2}\theta}\right)-qRE_{0}\cos\theta\ .\end{eqnarray*}\]Now this formula is extremely useful. Using it we can entirely understand the movement of the molecule. Let us for example set \(p_{\varphi}=0\) for the moment. We find that \(\dot{\varphi}=0\) and\[\begin{eqnarray*}\mathrm{E}\left(p_{\varphi}=0\right) & = & \frac{mR^{2}}{2}\dot{\theta}^{2}-qRE_{0}\cos\theta
\end{eqnarray*}\]which is an oscillator equation for small \(\theta\). So, for small energies the molecule oscillates back and forth in \(\theta\).
But what about a non-vanishing angular momentum \(p_{\varphi}\neq0\)? Here one can look at the "effective potential''. This concept originates from celestial mechanics. There, one is for example regarding the orbital motion of two bodies under the mutual influence of gravitation. If this motion has a certain angular momentum, the bodies may never collide.
In our case, the situation is just like that. Rearranging terms we find\[\begin{eqnarray*}\frac{mR^{2}}{2}\dot{\theta}^{2} & = & \mathrm{E}-\frac{p_{\varphi}^{2}}{2mR^{2}\sin^{2}\theta}+qRE_{0}\cos\theta\\& \equiv & \mathrm{E}-V_{\mathrm{eff}}\left(\theta\right)\ .\end{eqnarray*}\]Let us take some characteristic quantities for molecular systems to look at \(V_{\mathrm{eff}}\): \(p_{\varphi}=\hbar=6.63\times10^{-34}\text{Js}\) (Planck constant), \(m=m_{p}=1.67\times10^{-27}\text{kg}\) (proton mass), \(R=a_{0}=5.29\times10^{-10}\text{m}\) (Bohr radius) and \(q=e=1.60\times10^{-19}\text{C}\) (elementary charge). Then,\[\begin{eqnarray*}V_{\mathrm{eff}}\left(\theta\right) & = & \frac{\hbar^{2}}{2m_{p}a_{0}^{2}\sin^{2}\theta}-ea_{0}E_{0}\cos\theta\\& \approx & \frac{1.19\times10^{-21}\text{J}}{\sin^{2}\theta}-8.48\times10^{-30}\text{A s m}\cdot E_{0}\cos\theta\end{eqnarray*}\] and we can see that an electric field has to be as big as \(10^{9\dots10} \text{V/m}\) to actually have an effect!

In the figure you can see how \(V_{\mathrm{eff}}\) changes for different \(E_{0}\) from \(10^{10}\text{V/m}\) (blue curve) to \(6\times10^{10}\text{V/m}\) (green). At different energies \(\mathrm{E}\) the molecule oscillates in a larger \(\theta\)-range. But similarly to the case of orbital motion, the points at \(\theta=0\) or \(\pi\) are never reached since \(p_{\varphi}\neq0\). Along with the angular movement in \(\varphi\), we can conclude that the molecule is performing a kind of precession.
We have seen that the details of the motion depend on the balance between the conserved quantities and the strength of the applied electric field.