Locksets for 1 1/4" Thick Doors?

All of my interior doors are 1 1/4" thick and have standard but old (35 yr) locksets. I would like to replace them but I can find no locksets that fit 1 1/4" doors - (1 3/8") is the minimum). I suppose I could shim them but that would probably look crappy. I found mobile home locksets that would work but the hole for the latch is 7/8" and mine are drilled standard 1". Shortening the center shaft of a new lockset as well as the sleeves that accept the 2 mounting screws might work but woud be tedious. Any other ideas/solutions would be greatly appreciated.

Paul