ok has some miles on her.
When the mower is under a load, the engine starts stumbling-cutting out
Typically a main jet is clogged or a dead cylinder if you have two. Make sure you have a solid flow of gas AT the carb inlet.
I swapped the carburetor with the original, which I had replaced to troubleshoot a hard starting problem. (It didn't fix that issue, so I believe the carb is good.)
From 2006, that carb is probably plugged up.
I swapped the carburetor with the original, which I had replaced to troubleshoot a hard starting problem.
So are these carbs OEM Kawasaki parts? Or Chinese Scamazon/Fleabay?
While it is stumbling, there is no evidence of smoke indicating running rich
What do the plugs look like? Put pictures up here so we can see.
My next step is to start throwing money at it, ignition modules, etc.
NO your next step is TEST WHAT YOU HAVE. Here is what you need. Air, fuel, spark "at the right time" and compression.
So you need to check;
1.Pull air filter for TESTING only. See if she will run under load without the filter for a BRIEF PERIOD OF TIME. No change, check air off your list unless the valves are fully out of whack.
2.Pull fuel line AT carb inlet. Drain into a glass jar. Should have a fat steady flow.
3.Pull each plug. Ground them out on a cooling fin or something off the block. You can get those cheap Christmas light kind for each plug. Don't like those myself. Crank engine looking for spark.
4.Pull top engine shroud. Look at flywheel key. Should be in the slot and not stripped out or missing. Take a picture so we can see.
5.Compression test, watch youtube videos if you don't know what to do here.
Good maintenance items are carb, plug cleanings and valve clearance inspections. Clean your cooling fins YEARLY or more often if you bag or scalp.