It depends.
In the LDs with a lower level of awareness it feels a bit like "mild anesthesia" just like you said, but I never noticed a delay between stimulus and sensation... I shall pay close attention to that in my next LD. However in the LDs with a higher level of awareness it all feels like it does when I'm awake - maybe even a bit more intensive. Especially the wind on my skin when I'm flying or running feels extremly real. Also when I concentrate on a sensation (for instance touching the floor in order to stabilise) it feels very real and I can feel the temperature as well as the texture.