Main Question: What should one look for on a sport hybrid if the primary use will be commuting and riding around 10 - 20 miles a day for exercise?
Motivation: I was a resident of Chicago (which was absolutely beautiful for biking along the shore and such) and I had a 1977 Trek road bike that I loved. Now I live around the New Orleans area and the roads are absolutely terrible to bike on; I have replaced tires, tubes, and rim tape a number of times, but it never quite works; it's getting to the point where I'm worried about going farther distances because I'd be stuck there if my tube popped (and this has, in fact, happened a few times...). I went to my LBS and they told me that I should think about getting a sport hybrid --- and if this term is not standard, it seems like a hybrid which doesn't have shocks.
I am reluctant to buy one because I don't know all that much about bikes yet and I'm not sure what to look for in this type of bike; moreover, I am used to drop handlebars which are not standard in sport hybrids. Does anyone have any experience with these kinds of bikes or from going from a road bike to a sport hybrid? Are these kinds of bikes good for commuting to work ( < 5 miles) and for longer exercise rides (10 - 20 miles), or would a different bicycle be more fitted for these purposes?