When you mean towards the driver, you’re referring to pushing it towards the back of the mower correct? I just tried tapping/gently pushing all the linkages towards the back of the mower - no change. Interestingly enough, I got a coolant leak now. Any thoughts why it randomly popped a coolant leak by the radiator?
Another thing to note is that it cranked for a long time after the spark plug change to get started. And just as it was catching, I think there were a couple of puffs of white smoke (couldn’t tell it was dark and overcast outside), which then reverted back to the black smoke and the rough idling.
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Driver's side, as in a car (sorry that wasnt clear)... if you are looking toward the front of the mower... you would kneel down by the left, front wheel... find that metal plate where throttle/choke attach... and push on all the linkage toward the right side of the machine.
You said you changed the carb... did you get the governor adjusted/ put back correctly?
As for the coolant leak... probably radiator needs some work if it's not a head gasket (which sounds like some of your symptoms could resemble that but not the black smoke... and if your oil isn't milky then... should be fine on that front.)
If it's the radiator... find a good radiator shop and have it fixed. People who cannibalize parts from non functional machines have the parts (like a radiator)... but you'll overpay big time.
I highly doubt Kawasaki still sells this radiator. And if they did it would be big money anyway.
IF you don't have a radiator shop nearby AND there's not a reasonable one listed on ebay/ craigslist/ marketplace... I would just look for a complete engine from a used LX 178/ 188... and I *think* some other JD models used the FD series engines (like the early UTV & commercial front mower deck models... might could use those radiators too but not sure.)
Top end on these things are always "gunky". I think they came with stock 160 degree thermostat... which doesn't help with the rich as a fat pig jetting. So yeah... they are messy up top... sounds normal your description.
John Deere did eventually offer a 180 degree thermostat... I actually think there was a bulletin on it. I think they still sell it.