If you want to go that route, you may find it easier to check your logic analyzer dump side-by-side with the debugger in an emulator (e.g. VICE). Break on the first instruction and then go step-by-step. (Just need to make sure you have the same ROM version, but that shouldn't be too hard.)

However, your problem kind of sounds like a power supply problem. So using a logic analyzer will maybe just produce a different result every time. So maybe check the 12V and 5V rails on an oscilloscope while turning on the computer. (Or maybe it's a problem with the reset circuit, etc.)