I can see why you are frustrated.
You have spent a lot of money replacing items that are not related to your problem with little return.
The solution to every problem is to first work out the cause of the problem then rectify it.
IF the engine will only start & run with the choke on then you have a situation where the air:fuel ratio is way too lean.
Two reasons for this
1) too much air entering after the fuel is mixed in
2) not enough fuel in the mix in the first place.
The first is caused by an air leak between the carb & the inlet so you test for it by starting the engine then saturating the manifold region with WD 40 or similar using a trigger pack, not a spray can.
If you get clouds of white smoke then you have a leak.
The second, not enough fuel can have a lot of causes, from plain bad fuel through to a damaged carb.
As your problem is fairly well immediate, the easiest way to confirm he fuel delivery is to remove the fuel line at the carb , put it into a coffee cup sized jar and crank the engine .
If the system is good it should fill the jar in around a minute .
have a good look at what is in the jar, checking for water &/or dirt.
Now you said you replaced the solenoid
Which one?
Under the carb or on the starter ?
If the former did you check it was working ?
If the latter then the sales persons pension fund thanks you greatly.
If the fuel supply test mentioned earlier is good then you turn your attention to the carb
AS you did not care to tell us what engine you have fitted to your John Deer we can not tell you how to clean your carb.
However you will find comprehensive instructions on cleaning carbs here
http://outdoorpowerinfo.com/ in the repairs index.
John Deer produce a very comprehensive repair manual for your mower, including the engine and the cost of it is less than what you have just spent replacing perfectly good parts