If you think that the rod is a 6 wt., you can try loading new backing and a 6 wt. line. I was told by an Orvis rep that you can go up or down one wt. in a fly line from what the rod is rated for. For example, I have a 6 wt rod with a med flex tip, and I will switch back & forth between a 5 wt. and a 6 wt. line depending upon the circumstances. I would try it and see how it casts. IMHO.
For a replacement cork handle, I would look for websites from rod building companies. It may involve some trial & error to find one. Have you done a search on this Forum? With all of the knowledge and experience of the members, I'm sure someone can help you out.