What you are describing sounds like one or both of your rear brake calipers (assuming that the emergency brake acts on the rear brakes on that car) is sticking or that the emergency brake is not releasing fully. If you put your hand close to the rear wheels after driving you'll probably notice that one or both of the wheels is noticeably hot – maybe even hot enough to burn you.
When the caliper sticks it causes the brakes to drag and it this also means that emergency brake doesn't have anything to push against. That's why the handle feels like it isn't doing anything. Another explanation of the problem could be something causing the emergency brake to hang up – one possibility might be water getting into he brake cable and freezing. Do you live in a cold climate?
You're right that a new car shouldn't have problems like this, but sometimes it happens and that's why there is a warranty. The dealer (and Honda) want to fix this for you – most people understand that out of the millions of cars sold each year that some will have problems and the industry knows that if they are responsible and proactive in solving the problems that that will outweigh the fact that there was a problem in most people's mind.
So take the car in and get it fixed, but also ask the for an explanation of why the problem happened – it might be hard to track down if the problem is weather related, but they should be able to tell you what they checked and what they found or didn't find. Try to make sense of what you're told and you can bring it back here for an explanation and/or sanity check.