Chanel Resistance in a saturated MOSFET
Interesting question.
I simulated for a year electronic semiconductor devices.

Channel resistivity is a 2D function: ρ(x,y).
ρ(x,y) changes as VDS change.

To answer your question:
Does this mean that the resistivity near the drain have the highest value?
Yes, it does.
Is the dissipated power mostly generated near the drain?
Yes, it is.

The external circuit will see the overall channel resistance R which is the 2D integral of ρ(x,y).
You increase the channel width W, you reduce R.
The electric field vector E(x,y) and the current density J(x,y) are 2D functions as well.