Assigning 1H NMR spectrum and predicting peak coupling patterns for 2H-1-benzopyran-2-one
Don't you have an integral for this spectrum? This would show you clearly that you have two sets of overlapping signals, just as you have surmised.

Looking at the numbering for your molecule described:



You are correct with the assignment for the vinyl protons. Proton E will be strongly downfield shifted, and F only moderately shifted from the base double bond chemical shift of 5.25. These values could be determined using additive tables found in lots of NMR textbooks.

The proton shifts for the aromatic ring can also be roughly calculated using additive tables, but what you have are indeed overlapping signals; Protons A and C are at ~7.3 and protons B and D are at ~7.55 ppm.

To predict the splitting patterns you anticipate for the aromatic system, consider first the general couplings you'd expect:


ortho coupling ~7-8 Hz
meta coupling ~2-3 Hz
para coupling ~1Hz or less


Let's ignore the para couplings for simplicity - they are not often seen.

So, 


Proton A will have one ortho coupling and one meta coupling, so a will be a dd
Proton B will have two ortho couplings and one meta coupling, so will be a ddd
Proton C will have two ortho couplings and one meta coupling, so will be a ddd
Proton D will have one ortho couplings and one meta coupling, so will be a dd


Let's now build in double bond :


Protons E and F have a cis coupling (~10 Hz). 
Proton F will appear as a doublet, but proton E is likely to have a homoallylic type coupling to proton D (4J), splitting both of these signals to an additional doublet with a coupling of about 1-2 Hz. This means that proton D may be a ddd, and proton E a dd if this coupling is observable.


So that should be the expected splitting pattern from this molecule. Now to justify the appearance of the peaks. Overlapping signals can often make things difficult to interpret. Without knowing what frequency this spectrum was recorded at, you also need to consider whether there are second order effects starting to manifest, certainly with the meta couplings. There will definitely be tenting of the peaks giving rise to distorted peak intensities from what would normally be expected. 

So, I wouldn't be devoting too much time to the interpretation of overlapping signals like these. One thing I will say is that if you label the peaks as (say) a ddd, then you should be able to report all couplings for each d. Otherwise, if you cannot report couplings (and this means couplings, not splittings) report them as a multiplet (or in this case overlapping multiplets).

Complex splitting patterns of second order and overlapping signals can be analysed and simulated, but it takes a lot of work.You can try drawing out a splitting diagram for each signal using the approximate couplings for ortho and meta couplings, and then start overlapping through constructive addition to see what you get. It gets very complicated very quickly.