Regarding the stability of the Schrödinger equation, the main result is that the propagator is a unitary transformation, which therefore preserves inner products. In particular, if $|\varphi(t)\rangle$ and $|\psi(t)\rangle$ are arbitrary solutions of the time-dependent Schrödinger equation for any arbitrary hermitian (possibly) time-dependent hamiltonian $H(t)$, then
$$\langle\varphi(t)|\psi(t)\rangle = \langle\varphi(0)|\psi(0)\rangle$$
for all times $t$. In particular, it means that if you perturb your initial condition $|\psi(0)\rangle$ into, say, $|\tilde\psi(0)\rangle = a|\psi(0)\rangle+|\chi(0)\rangle$, where $\langle\chi(0)|\psi(0)\rangle = 0$ and $a^2+\langle\chi(0)|\chi(0)\rangle = 1$, then the solution will retain that form: you'll have $|\tilde\psi(t)\rangle = a|\psi(t)\rangle+|\chi(t)\rangle$ for all time, where the perturbation stays orthogonal, $\langle\chi(t)|\psi(t)\rangle = 0$ and the projection on the original solution $a^2 \equiv 1-\langle\chi(t)|\chi(t)\rangle$ stays constant.
Regarding the existence of oscillatory solutions in both theory and experiment, I gave a good description in Is there oscillating charge in a hydrogen atom?. Generally, if you prepare any superposition of two eigenstates of the Schrödinger equation you will get a periodic solution. If you prepare a superposition of a finite number, you'll typically get a quasi-periodic solution (unless the energy differences happen to be exactly commensurate, which is hard to arrange, in which case you'll have an exactly periodic solution). If you've got a superposition of a continuum of eigenstates, then generically you will have no recurrences at all.
Regarding your statement
Yet, oscillatory solutions are in some sense time independent ("stationary") and we should, maybe, think of real atoms as being at oscillatory states.
that's not really how you think should think about it. Oscillatory solutions are not time-independent - they are oscillatory, period, and it doesn't really help if you try to deform the notion of stationarity to include those cases. Stationary states are really stationary: wait any arbitrary time $t$, and you get an equivalent state (i.e. equal up to an irrelevant global phase). This is not the case with oscillatory solutions.
And finally, if you want to speak about "real atoms" - that completely depends on the situation. The TISE is a tool, as are the hamiltonian's eigenstates. Sometimes real atoms are in eigenstates, sometimes they're not. Sometimes they're in pure states, sometimes they're in mixed states. It depends what kind of experiment you're describing. Trying to speak about "real atoms" in a generic way will swallow you up whole because of the wide array of different real-world situations those atoms can find themselves in.
