I recommend going cold turkey! I went through this with my 3rd child, my older two didn't use a binky past 8 months old- their choice- but my 3rd was a binky boy. I did the same thing. Hunting under cribs and behind dressers in the middle of the night. Panicking when I couldn't find one.....
Cold turkey is easier than you might think. It's always harder on the parents than the kids to take away their favorite things. I nixed the bottles as soon as my 3rd turned one (I was SO over washing bottles at that point as my children are so close in age) and wasn't about to do the Binky Scavenger Hunt any longer. My son was 18 months when I took his binkies away. One morning, I just said, "No more!" and threw them all away.
You may lose a little bit of sleep at first, but how much are you losing now? The first night, my son asked for his binky and offered him a stuffed toy to hold instead. He surprisingly agreed and it seemed as if he had all but forgotten about his binky in two days.
If you're not ready to toss the binky yet (I totally understand!!) they do make binky clips to attach them to their clothing. My only problem with those is you have to make sure you clip them far enough down that the binky JUST reaches their mouth. Otherwise, I worry about them getting strangled by the ribbon thing it attaches to.