I really don't agree that "fourmis dans les jambes" & "engourdi" are the same.
It happens a lot to me that after being sitting in my chair for too long, when I try to get up, I no longer feel my foot, which is a weird sensation. When it happens, naturally I say:
"Je ne sens plus mon pied."
"Mon pied est engourdi" is correct but more formal and sounds a bit stitled to me. Naturally I would never use it.
(= My foot has fallen asleep)
And for "is falling asleep". "Je commence à ne plus sentir mon pied."
"Mon pied s'engourdit."