Thanks!
Sticky to me means they won't fan hardly at all (they feel stuck together, clumps)
It will largely depend on what's causing them to stick together - common grime found on the edges you might be able to work with, but PB&J stains would be a totally different ball game.
With modern cards, I've taken a DRY paper towel and pushed hard along the edges in an outwardly direction - it's helped to remove some of the dirt and improved performance.
Bear in mind as well that when it comes to old cards, they weren't made to the same standards as a modern deck. The finishes weren't as good, they weren't always embossed, they're often found used... I wouldn't let performance issues stop me from getting a vintage deck if it was something rare or beautiful.
There's a trick that I know some sports cards traders have used for removing grime on the edges of cards, though it's something you'd want to test first. A very light application of lighter fluid rubbed gently with a cotton swab or a paper towel might get the dirt off more effectively - but you want to do it in a ventilated area away from open flame and you want to test it on a card of the same type that you don't care about first before applying it on an entire deck. Sports cards don't have the same coatings modern playing cards do, so modern cards would likely be stripped of their plastic coatings by this method of cleaning. However, older playing cards might be less negatively impacted because of the differences in coatings used at the time - you'd really want to test it, perhaps on a joker, an ad card or even a torn/damaged card, to see if it will ruin the cards; I can't emphasize that enough.