My dad has a 99 LeSabre and it's been doing the same thing for years. Drives fine, and runs perfect, even WOT downshift into 1st at 20mph. But mashing it more than half throttle from a stop, and the car shakes hard and shimmies, like the tires have broken loose and are wheel hopping, until you let off the gas. It'll do it hot or cold, and consistently.
I'm thinking it's caused by the input clutches cycling in and out rapidly, caused by low initial trans oil pressure, because it never slips or shakes after it has had RPM's on it, i.e. the WOT downshift into 1 doesn't slip.
There is an idea I'm going to try. Like a brake stall test. Hold the brakes down hard, then give it maybe 1/4 throttle for a few seconds, let the pressure in the trans build, then mash it and let off the brakes.. If it doesn't shake on the mash, I'm thinking that would kind of verify the low pressure..
I'm guessing there is a partially blown out gasket somewhere on the Valve body or Channel plate. Input piston seals from the factory are weak on 4t65e's also. That makes sense because the issue is "static", not getting worse over years.
Fixing any of that basically requires pulling the tranny out, and if ya got the damn thing out, might as well at least do a light rebuild (gaskets, seals, frictions etc) That's what anyone who has done it before, is thinking. Unless you are just going after a known bad solenoid on a low mile trans..., well I'd be close to suicidal if I went thru to time and hassle to pull the trans to replace a single part, and have something else give out 500 miles later. Drive it till it dies.