Sounds like your technique is good....speed, gas on, weight rearward. Agree with Planalp and Scrapper but will add the following if you have not already done so.
1) Check the steering head bearings for looseness / binding / tight spots. (free).
2) A fork brace may help. (semi-cheap).
3) Steering stabilizer. (expensive). I've never actually seen one on a KLR.
Is your bike loaded the same as the last time you rode on this?
When you say "Like riding on marbles on top of concrete" I know what you mean. The old stinkter gets as tight as you wish the bike was.
Gray-haired riders don’t get that way from pure luck.