Well, I tie my own and put a couple wraps of lead wire on the center of the hook shank. Not enough to really drag it to the bottom but enough that it sinks a little.
I've also tied them (and bought them) unweighted. The unweighted ones seem to do better for me in slower current with just a lazy sinking drift.
I pretty much just chuck them upstream and a little across and let them drift on by me. I've caught a lot of fish (trout, bluegill and bass) at the end of the drift as they flutter around in the current.
I think they look like some kind of inch worm or caterpiller to the fish so I try to imagine what one of those critters floating in the water would look like and fish it accordingly.
I love the weenies. It's a great and simple fly both to tie and to fish.
Hope this helps.