Won't start. Battery fully charged. Clicking. Everything else OK.

A "dead short" in the starting system would cause lots of smoke and maybe flames so I don't think there's a short circuit directly to ground. I think you mean that there's no starter action when you turn the key to START. I suspect that the starter is failing. Check that you have battery voltage at the big starter cable, and battery voltage at the smaller START wire at the starter when your assistant turns the key to START. If so, and the starter doesn't run, then it's faulty. Don't assume that the starter has to act just like the one you replaced, failures come in many varieties. If you want to find a Caddy specialist, here's our directory link for you: https://repairpal.com/cadillac-repair-in-fairview-tennessee