A wicked shimmy when towing generally comes from one of two things.
One - your steering stabilizer is worn out.
Two - your alignment is off. You need enough positive caster to keep the wheels going straight. Some cars are a bit tight on caster and when you add a tow bar base plate it adds extra weight to the front end (same as with worn springs) and the caster goes to zero or even negative caster. Then the vehicle drives like one of those shopping carts with the worn out front casters.
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Mark & Leann Quasius
2016 Cornerstone 45A
2020 Jeep Gladiator Rubicon
2021 Jeep Wrangler Rubicon
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