As others have said, a burned connector is common, and should be replaced whenever possible. But a burned connector doesn't directly mean a shorted line. If the connector burned enough, there's simply no more contact or the resistance is way too high for current to flow. Measuring to see if there is a short is super easy. Just locate one of the bulbs that doesn't light and measure continuity between its 2 tabs. All the bulbs are in parallel on the string.

The fuse didn't blow, because there probably isn't a short, but rather an open circuit because of the charred connector. You should inspect all connectors while you are at it.. The input voltage might be charred as well and might eventually cause resets.