Body shakes left and right when accelerating. Due to the sway bar link?

What is your question?

When I accelerate to 35 to 40mph, especially when climbing the ramp to the highway , the car body shakes left and right, but disappear when I hit 60mph. Checked tire and wheel balancing at discount tire, the problem persists.

What makes this problem better or worse?

It‘s worse when I am climbing the ramp to highways. But shaking disappears when speed goes higher like 60mph.

How long has your 2008 Lexus ES350 had this problem?

Since I got this used car.