You don't need a new neck - or a new banjo. All you have to do is set the bridge.
The wig tape is a bad idea. I would be willing to be money that the bridge is in the wrong place - banjo players don't glue, tape or otherwize stick the bridge down on the head. The bridge is only held in place by string tension.
So before you buy anything watch the banjo setup workshop I filmed last week. http://youtube.com/watch?v=yiilZ-PD1uQ Pay particular attention to how I set the bridge.