As for the shaking, check your rear tire pressures. These things are sensitive to having the rear tires at different pressures, makes it shake and wobble. Also check your front tire pressure to. Check the bearing for excessive play, and also check the head bearings to see if your steering is loose. What i usually do is get on the trike and apply just the front brake then rock the trike front to back to see if the steering wobbles. Sometimes you can just take it apart and clean and grease and retorque and it will take the slop out, worst case the bearings are shot and you will have to buy a set of roller bearings from all balls to fix that.
For the hanging throttle problem i would just spend the money and get a new cable. Oh and make sure the needle in the slide is on the right clip position! In that one picture you posted it looks like the needle is down pretty far, if the needle is bottoming out before the slide can go down all the way you will be adding extra air at idle and she will run lean and rev up a bit, been there done that.
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1984 Honda ATC200ES "Big Red"
1982 ATC200E "Hondie"
1988 TRX300FW "Project Quad" Still in progress....