Addressing a number of points in this one:
I'd checked out the Lockport family, and I agree that they're not the right ones.
I've been digging around in New Orleans and have gotten some interesting leads. It appears that John F (husband of Alice) died somewhere around 1875. The city directory shows him listed as a grocer until 1875. The 1876 and 1877 directories have no listing for him, but the 1878 directory shows "Mrs Alice Crane" as a grocer at the same location. The 1880 directory is the first one in which she's listed as the widow of John F.
Of note is that there's a John F Crane in the 1866 directory listed as a blacksmith. I think this might be our John F, based on the location given (Algiers).
There are two William R Cranes in the directories; one is apparently a William Rufus Crane and, as far as I can tell, isn't related. The other one is probably the son of John F and Alice (his occupation is given variously as a blacksmith and warehouseman). The other John F Crane (the son) appears to have died in 1890.
I haven't been able to locate the civil case David mentioned, but I did find a mention of the succession papers (aka probate) being opened after Alice's death in 1903.
And I have done my DNA testing with Ancestry (and 23andMe and FTDNA and My Heritage--thanks to the ability to upload existing data, it's been less expensive than it looks), but I haven't gone down the Crane road yet--thanks for the reminder.
And I can't begin to express my appreciation for everyone on this board!