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?