The headlamp washers on my 2016 LS 460 do not fully retract / hide back into the front bumper after use.
I have to press on them at the same time which pushes out some washer fluid and they then retract back.
Is this by design or is something broken or clogged that I can fix myself?
No, there is nothing physically blocking them other than the washer fluid pressure.
See attached picture...

 

These washers being pushed up by the fluid pressure. Nozzle could be partially clogged up and not releasing fluid pressure all the way, which is what probably holding it extended.