Did you obtain copies of the death certificates for Lura and Lee?They should have some info, names and birth locations of parents, and SS number for Lee.Also, I'd obtain the one on Cobb Pickering, he really looks like a match, and his parents would be Lura's parents.You have to get sneaky sometimes, and go up the family tree on another limb.
Also, look for obituaries on Lura & Lee, sometimes they contain info not included on death certificates.If none of these gives you Lee's birth location, obtain a copy of his application for SS card, he would have filled it out, and it should show where he was born. (Let's hope it says more then just OK)Hang in there, one of the clues will pan out.
By the way, I looked in TX Dth Idx Online and SSDI for a Lee Elliott d 1984, and didn't find a "exact" match.But, there is a Millage Lee Elliott b 8 Sep 1885, d 9-28-1984 in Van Zandt Co, TX, last residence Canton, Van Zandt Co, TX, and SS card issued in TX.
If that is your man, you will need to use his full name when obtaining death record.