I figured out that the problem of my calculation actually lies in the incorrect use of 4-vector representation for the pair interaction, which was essentially due to the incorrect use of Feynman rules. We can clear the confusion by going back to look at the contractions of the time-ordered operator products. The natural form of the potential, in the case of potential scattering, or the pair interaction, in the case of electron-electron interaction, when put to interaction representation, does not explicitly contain time $t$. Therefore, it is at beginning more natural to directly work with usual 3-vectors. For the Hartree-Fock term, it is the lowest order correction coming from the following expression:

$$\langle \phi| T\int dtd\textbf{x}_1d\textbf{x}_2 \psi^{\dagger}(\textbf{x}_1,t)\psi^{\dagger}(\textbf{x}_2,t)V(\textbf{x}_1-\textbf{x}_2)\psi(\textbf{x}_2,t)\psi(\textbf{x}_1,t) |\phi\rangle,$$

which after contraction will be something like

$$\int dtd\textbf{x}_1d\textbf{x}_2V(\textbf{x}_1-\textbf{x}_2)(G^2(0,0^-)+G(\textbf{x}_1-\textbf{x}_2,0^-)G(\textbf{x}_2-\textbf{x}_1,0^-))$$

which after being put into momentum space would be

$$\int dt d\textbf{p}_1dw_1d\textbf{p}_2dw_2dqe^{iw_10^+}e^{iw_20^+}G(\textbf{p}_1,w_1)G(\textbf{p}_2,w_2)V(\textbf{q})[\delta^2(\textbf{q})+\delta^2(\textbf{p}_1-\textbf{p}_2-\textbf{q})]$$

$$=TV\int d\textbf{p}_1dw_1d\textbf{p}_2dw_2e^{iw_10^+}e^{iw_20^+}G(\textbf{p}_1,w_1)G(\textbf{p}_2,w_2)[V(\textbf{q}=0)+V(\textbf{p}_1-\textbf{p}_2)]$$

From this expression we can clearly see why propagator part can be put compactly as 4-vectors, while the interaction part cannot.

What I have been shown here are for the Hartree-Fock diagram without loose ends. But with the same correct idea, the expression for the diagrams in the original question can be readily written down.

**Painful lesson learnt**: for people trying to get hands on Feynman diagrams, try not to become overly obsessed with the fancy diagrams before one has done sufficiently number of pages of calculations with the conventional "dumb" way of calculating correlation functions. The diagrams work very nicely only when we know exactly what they represent.