If I am driving on a smooth road I have no problems, but if I hit just a small bump or pot hole there is a damped shake of about 1 to 2 seconds and then reverts to normal This sounds to me like front struts, but they only have 100K on them. My SC went almost 200K before they needed replacement.
Does this sound like struts to you?

 

My guess is yes. Have the car checked for suspension safety, however: ball joints, control arms, etc. You don't want a wheel coming off. It happened to me years ago (really, it seized on the highway, but the wheel really was nearly detached) due to bearings that had no grease even after I asked for them to be repacked at the prior brake job. Unpleasant, and modern cars have sealed bearings, but my point is don't risk it. I have air susp with 165k miles, and it's OK but I think new struts would improve the ride. Still, an unusual shake could easily be just the struts.
,
I just spent small fortune on my Cadillac XTS to get new struts ("magnetic ride control" so fronts retail for $800 each on the parts, and this was not a place I could bring my own). . Car was groaning over bumps like a screen door and various clunks on low speed turns. The dealer 1 year ago said it was not struts, had no idea. The indie mechanic this year showed me tire wear, etc. (Inner edge of tire tread looked... mottled. awful). New struts and shocks (all 4 wheels) and car is riding much better: not smoother, but no more bad sounds, etc.. Tires were low on tread anyway, but the struts/alignment clearly damaged them worse. I got new tire immediately. Just be glad you don't have air suspension on your LS. I read about replacing with non-air on this forum, but still not confident if I ever need to that I won't lose my AFS (swiveling headlights) if I pull fuses for the air susp...
On the Cadillac 50k is normal for leaking, etc. So 100k for any struts is a lot, even if your other car made it further. But more importantly, check out the rest if the suspension for safety and your tires for hidden wear on the inside portion of the tread.

 

I had similar issue on 04 with 100K, had pretty dramatic steering wheel shake mainly when hitting a bump at highway speeds. According to the Lexus service history PO had been into the dealer frequently for some front end issue...many alignments were done and Lower Control bushings replaced, not sure they ever found the problem. I swapped out front struts, no change. Then after trolling this forum found perhaps Steering Rack bushings? Swapped them out 100% fixed the issue,.feels like driving on rails, a new car again.

Part numbers, 2 per side

45516-50030 Drivers side
45516-50040 Pass side

Lexus does not sell these for 2004-06 so you have to look under 2001-03 to find them...but they do fit all years.

Pretty easy DIY there are pics on the forum for the process.