NMR: Most deshielded protons (electron withdrawal)
I don't believe there is anything really wrong with this question. Yes, both molecules have multiple signals, but one of them contains a signal which corresponds to it being the most deshielded. In fact, one of these, molecule F, contains the most deshielded signal AND the most shielded signal.

"In any case it's clear that the question is asking about which molecule has the highest degree of electron withdrawal ..." not necessarily so. The correlation between electron withdrawing groups and shielding/deshielding is a strong rule of thumb, but can become complicated with aromatic systems and competing electronegativities of substituents. In any case, you need to be looking at parts of the molecule; shielding is quite a localised effect, and as I said above, molecule F actually has the most shielded AND deshielded protons.

In any case, if you consider the electronegativity of oxygen; a proton attached to this will be strongly deshielded. Stick them on an aromatic ring para to a carbonyl group, and you will generate a resonance system like below:



In this case, the phenolic proton is by far the most deshielded of all protons in the molecules shown; I'd expect a chemical shift of over 10ppm. In response to some of the comments about -OH signals being over the place; it is important not to confuse raw chemical shift and averaged chemical shift resulting from chemical exchange with other labile protons, such as the (often observed) exchange of phenol protons with water. In these cases, the resulting chemical shift depends on the relative concentrations of the exchanging species, the exchange rate, and the original chemical shifts of the non-exchanging species.