You can try to get the wax out of the lining by first ironing it between layers of newspaper and then sending it to the dry cleaners. This is what we used to do with batique clothing where we had to get all the wax out. You use enough layers of newspaper so that the iron can get it hot but the wax won't leak onto the iron (actually best to use an old iron.) You iron it again and again, discarding the waxy newspaper and replacing it with more as you go. When you have gotten all the wax out that you can, the dry cleaner should take care of the rest. This used to work for us on cotton muslin and jersey fabric. Then you can get the lining repaired and re-wax the coat. I would try it; you apparently have nothing to loose.