I think the three bond coupling would change your assignment.
If you look at proton c, you see a single proton being split by two protons b in identical magnetic environments. This would give you a triplet. Looking out an additional carbon, you could see that it could be further split by two additional protons in identical magnetic environments - each peak of the triplet caused by b would be further split into a triplet by a. Being three bond coupling, the coupling constants a-c would be far smaller than b-c, and you would see a clear triplet of triplets.
Consider proton b - it is being split into a doublet by the adjacent a and a doublet by the adjacent c, but since the splitting constants are approximately the same for a and c (both being aromatic protons the same distance away), you get something that looks more like a triplet, where one peak from each large doublet lie almost on top of each other. There is additional splitting from the other b proton, and a very tiny amount of splitting from the other a proton. So, something that looks like a very messy triplet.
Finally, proton a - it is split by a single adjacent proton b to give a doublet. That doublet is further split into doublets by both c and the opposite a, giving something which might be sort of a doublet of triplets, or possibly a doublet of very tight doublet-of-doublets. Finally, each of those peaks is double very slightly by the opposite b proton.
How does this line up with your spectrum?