I've made mine out of buckets with lids (kitty litter), and also used rope hooks. Mine are single rope tie-down hooks, rather than a double cleat. They've been sufficiently strong, I use mine mostly for heavy groceries. The only problem I've ever had is that they can bounce away from the rack at times. If you added my hook/bungee arrangement and kept your drawer pulls you should have a solid set-up.

For the noise made by the handles bouncing around, I tied a bandana around it. They still bounce but they don't make that rattle anymore. I can't help much with the buckets themselves making noise, around town groceries make noise but never for long. On tour we keep our clothes and sleeping bags in them. Perhaps you could line them with drawer liner?