John Rae sussed what had happened to both ships and their crews after searching for them in the 1850's and hearing directly from Inuit about survivors. His findings when published were given such harsh treatment by Mrs Franklin, via Charles Dickens that Rae never got the recognition he deserved either as an explorer or for finding the fate of the expedition.
Worth reading Fatal Passage by Ken McGoogan, if only for general background about Rae's search and the land and environment.
Looking forward to seeing what you come up with.