First off I have not replaced a ignition switch to lock cylinder rod so take this with a grain of salt.
However members say working on the steering column outside of the lock cylinder and ignition switch (Easy to remove.) is not trivial. IE - Replacing the rod or anything around it is fairly tricky. Most suggested after the fact the ignition switch or lock cylinder was very worn and there was play and it was hard to tell where the ignition switch was at.(Off, on, or start.)
As I understand there is the ability to adjust the ignition switch left or right to adjust it.(At least I have heard aftermarket ones do.). So set the switch so they key has to be turned a fair amount before it will move from on/run to start mode, but not so far the rod runs out of travel before it does,(Thus will not power the solenoid) and when you let off in the “start” position it returns the switch to “on/run”. If it does your rod is likely working.
You could test the small wire at the solenoid on the fender well that gets +12v when the key is in the start position and activates it. Unhook the small wire that gets +12v from the ignition switch. Set up a test light or voltage meter to see when it gets +12v. Do a truck off to start, start to key release to on. Do that cycle 15 times and watch after each release from start that power from the ignition switch is no longer going to the wire on the solenoid. No issue you shoukd be good to go.
Now for a month pay attention to the start cycle and listen for the starter to go off. (Or to hear something else that would suggest it stuck.) These trucks are loud and why many who have had this issue did not realize it. Other members admit to knowing they had a issue but forgot to remind a family member or friend who drove it to manually move the key back from start to on.
Lastly a lesson learned here monthly on this site. New and rebuilt parts often are bad out of the box sadly. Most members found the stuck starter issue without doing any work to the electrical system. But you did a lot of work but my point is you added a néw starter. (Likely a rebuild.) So it is possible your starter was defective and it simply stuck on due to a physical problem and did not retract. So it could of been related to a issue in the cab, or you got a defective starter.
If the voltage gauge in the truck works or if you have a aftermarket voltage guage in your truck, keep an eye on it after starting. My guess is when the starter is stuck,it iis generating over 14v.(Why it fries the battery cables, but perhaps it is ground out, not over charging. My truck only generatea about 13v till I come off idle, so a spike to 14-18v or a drop to 6v after starting would hint the starter might be stuck.
Let us know what you find and post some pics of the work.