I'm sorry, but you have to admit that's a pretty amusing problem. Maybe your truck has reached "old age" and hence tends to leave the turn signal on for long periods of time! :-laf
But seriously, these trucks have a computer for everything. When you click the turn signal switch it sends a high-logic (+12v) signal to the front control module, which then activates the signal. Even the clicking noise you hear is simulated (ever notice how the click and the lamp blinking are at different frequencies?).
That means that somehow the computer that controls the signals is getting a +12v signal from somewhere.
What, exactly did the dealer replace? There's a turn signal stalk, and a "multi function switch", which controls all functions of the signal stalk, plus the hazard lamps switch... is that what they replaced?
Here's the scary part: an examination of the serivice manual indicates that the entire instrument cluster (EVERYTHING) runs through a multiplexer and is connected via a PCI bus to the "Front Control Module" inside the integrated power module. That means 1 wire from the instrument cluster to the front control module. Based on that fact, I believe your front control module is faulty and needs to be replaced, since there's no real wiring harness between the switch and the front module.
For those with a service manual, the relevant drawings begin on page 8W-52-2. I would love to post copies of them here, but I think that's illegal.