Why do we need an old perturbation theory? There are two types of perturbation theory corresponding to explicit lorentz-covariance of amplitudes. 
The first one is called Rayleigh-Schrodinger perturbation theory. It is based on following ideas. Let's have Hamiltonian in a form $\hat{H} = \hat{H}_{0} +\hat{V}$, where there isn't dependence on time and the second summand is proportional to the small parameter $\lambda$. Also we know solutions of equation $ \hat{H}_{0}\Psi_{n}^{0} = E_{n}^{0}\Psi_{n}^{0}$, where the upper indice zero refers to the proportionality to the zero degree of $\lambda$. Then we can build the full solutions of our Hamiltonian by the expanding solution in a form $\Psi_{n} = \sum c_{mn}\Psi_{m}$, $\hat{H}\Psi_{n}= E_{n}\Psi_{n}$, where $E_{n} = E_{n}^{0} + E_{n}^{1} +...$, $c_{mn} = c_{mn}^{0} + c_{mn}^{1} +... $. After that we can find contributions of a "small" part of the Hamiltonian to the energies and expansion coefficients. 
For example, $c_{mn}^{1} = \frac{V_{mn}}{E_{n}^{0} - E_{m}^{0}}$, and etc. This result may be used even for relativistic processes (like in QED).
In the case when there is dependence on time in Hamiltonian we are not interested in energy of stationary state, so we want to find only the coefficients of expansion.
The second type is a theory based on s-matrix formalism which give explicitly lorentz-covariant amplitudes.
The first type is little harder and less convenient in compare with the second one. But there is the question: some authors like Weinberg says that we need the first type (he calls it "an old perturbation theory") because sometimes it helps to understand some singularities which are existed in the first type. Can someone make this statement clearer?
 A: I know only two kinds of perturbation theory: time-independent and time-dependent perturbation theory. 
Within non-relativistic quantum mechanics, time-independent perturbation theory is referred to as Rayleigh-Schroedinger perturbation theory. However, you can surely use the theory relativistically and so covariantly, if you wish. Say you need the energy shift given by the linear Stark effect, then you'll need this theory. This is maybe what you call the 'old theory'.
Time dependent perturbation theory is used mostly in scattering or decay processes, or when the hamiltonian explicitly depends on time. This is because you need to study the evolution of the system from an initial to a final point. This can be done both relativistically and non-relativistically. For a non-rel example, see the Rutherford cross section obtained in non-relativistic time-dependent perturbation theory (and thus non cavariant). When time-dependent perturbation theory is used within a relativistic covariant framework, it normally bears the name of S-matrix theory.
In short, I don't see any easy way of calculating a first order Stark shift in time-dependent perturbation theory. You'll have to (or at least most easily) use time-independent pert. theory, which in your notation is the old theory.
