I do not like large, new, chain hotels
You may have to go with international chains in Dresden, because that's really all there is in the Neumarkt area, which is the completely rebuilt "old town center". I didn't do a thorough survey, but all the hotels here appeared to be business and luxury class. I stayed at the Steigenberger der Saxe and found it very nice, if not exactly cheap. If you want something cheaper, you'll have to go further towards the Hauptbahnhof. The shopping promenade between the Hauptbahnhof and Neumarkt has several hotels, although they're all chains, mostly budget. Don't expect quaint, Old World boutique hotels, they're all modern (as are all of the surroundings).
Please also share any favorite local eaterys Depends on what you mean exactly by "local". Keep in mind that virtually no business in the rebuilt center is more than 10-20 years old. The best meal I had in Dresden was, surprisingly, at a Canadian-themed restaurant (Ontario-Canadian Steakhouse). I ate a so-so meal at Freiberger Schankhaus, a more traditional-ish Saxon restaurant next to the Steigenberger. I had a better traditional-ish Saxon meal at the Sächisch-Böhmisch Bierhaus-Altmarktkeller, which I think is right next to where they probably hold the Christmas market.