How can I prevent my old card sleeves from sticking together?
I'd say if your deck is even relatively valuable, I'd upgrade from the penny-sleeve treatment. Penny sleeves are fine for protecting bulk rares and draft decks but the premium stuff is usually a tighter, tougher sleeve with opaque backs (which matters if you want to do any serious competitive play). I would avoid Dragonshield Gloss though because the backs get scratched pretty easily from the pointy corners while shuffling. The Mattes are fine and actually real good. 

On the topic of cleaning and storage, here's a few suggestions. if you live in a very humid area, throwing one of those silica packets in your deck box can do quite a bit of work. To avoid collecting those weird little dirt clumps, using a play mat removes the dirty table element and will prolong their cleanliness for a bit (and plus they're nice to play on). If they do get dirty, rubbing alcohol will clean them up decently but after awhile, just replace them.