In Sakruari's Modern Physics, it's written that

$$H(t)|n;t\rangle =E_n(t)|n;t\rangle$$ simply noting that at any particular time $$t$$, the states, and eigenvalues may change. If we now look for general solutions to SE of the form $$i\hbar \partial_t|\alpha;t\rangle =H(t)|\alpha;t\rangle$$

then we can write $$|\alpha;t\rangle =\sum_nc_n(t)e^{i\theta_n t}|n;t\rangle$$ where $$\theta_n(t)=-\frac{1}{\hbar}\int_0^tE_n(t')dt'$$

But consider what I have done. $$i\hbar\partial_t|n;t\rangle =H(t)|n;t\rangle =E_n(t)|n;t\rangle \rightarrow |n;t\rangle =e^{i\theta_nt}|n;0\rangle$$ $$|\alpha ;t\rangle =\sum_nc_n(t)|n;t\rangle =\sum_n c_n(t)e^{i\theta_nt}|n;0\rangle$$ This is different from what is done above. What's wrong with my thinking?

$$H(t)|n;t\rangle =E_n(t)|n;t\rangle \rightarrow |n;t\rangle =e^{i\theta_nt}|n;0\rangle$$
The "evolution" of the basis states $$|n;t\rangle$$ is given by the requirement that it should be the $$n$$th eigenvalue of $$H$$ at time $$t$$. There is no requirement that this should match the evolution given by the Schrodinger equation. That is (if you will excuse the slightly butchered notation) $$| n;0\;(t)\rangle := U(t) | n;0\rangle \ne |n;t\rangle$$ where $$U(t)$$ is the time evolution operator.