L.L.
M., it is possible your son is hungry and wants to nurse during the night, not suck a binky.
A ten week old baby should nurse two to four times during the night, yes?
A baby cries because it is their only method of communication and usually if one cries when they should be sleeping something is amiss.
Soiled diaper, hunger, cramp, gas pains, ect ect.
A mother of a ten week old seldom gets good sleep.
My best advice to you is to sleep when your baby does, afternoon nap, morning nap, whatever it takes.
And nurse ( or bottle feed) your baby when he wakes at night.
Best wishes and God bless
Grandmother Lowell