What do you need distributions for? Your request is strange, PDEs are the fundamental application, the origin, and the main source of examples for distribution theory, so no surprise all the books on distributions after a while steer to PDEs.

Thus maybe my advice is misguided since I do not understand your needs. Anyway, in my opinion the best introduction to distributions is a nice little collection of exercises written by Claude Zuily some years ago (Problems in distributions, North Holland). If you finish it you will be familiar with all the basic theory and you'll be ready to delve into the intricacies, which can be challenging (see the first volume of Hormander, which is essentially a treatise on distributions, or the fear-inducing first volume of John Horvath with its fourteen different topologies on spaces in duality :)