I'm another LeSabre with an acceleration shake. It starts at 40-45 and fades by 60mph. Also, it is most prevalent when steering to the right, fading as I steer to the center and is gone on a left turn. The shake will stop when I get off the gas or if I kick it into passing gear ( 3000+ rpm ). First off I knew the suspension was worn so I replaced struts and shocks. Checked all the brake calipers to be sure there was none sticking /hanging up. I didn't like the front left wheel bearing, so I replaced both front bearings. Tires are new as are the brake rotors and pads. CV axles are tight with no slop at all. I considered the rack and pinion as it is present when steering to the right. Did have a leaking seal on the right side. Replaced that. I'm down to suspension bushings and the transmission. I'll be talking to the trans shop this week. I've been driving it in between the work done, and there have been subtle changes to the shake, but the basic problem is still there. I'll update after the trans shop and I will have the tire balance rechecked. One thought, I have been concerned about wear on the tires from this issue and so far I don't see anything undue. They were new last fall. Kinda makes me think it is in the rear suspension mounts?