Front end shimmy

I have a 2000 Grand Caravan with 220,000. I have had a severe shimmy on the front end for about 18 months. I had a friend replace the tires with used ones then. He noticed a bad tie rod on the right side. I had that replace and the wheels aligned. The shimmy was still there but not as bad. I had the right axle replaced a few months ago. It still shimmied. I took it back today to have it fixed again. They showed me that the right axle was fine but the yolk at the outer end wobbled when at idle speed. The left sway bar is broken. This shimmy usually is present at all speeds but worse when accelerating through 40 to 50 mph. It almost stops when you take your foot off the gas. What would cause this?

My feeling is that there is something going on that is creating stress and causing the failure of parts aside from the main problem. The shimmy you are experiencing I expect just gets worse as the related parts begin to fail.

Do this, jack up the front end so that both (or each if you have to do it singly) tires are off of the ground. Try to move each wheel vertically ,top and bottom, and horizontally, right and left. Use a quick motion and listen for a sound like a a “clunck”.
The wheel should not clunk or move with very little pressure for a short distance and then move under more firm pressure. A loose wheel such as this likely has failing ball-joints, which must be repaired as soon as possible.
While you are at it visually and manually inspect the front-end. Look for thin and rusted front-end components, especially the tie-rod ends. It’s not relevant to your problem but you need to check these things on an older car.

A constant shimmy, eliminating tire wear, is serious. Get it checked out and repaired.