That's the main symptom, and there is no known cure. I've heard that you can temporarily suppress the urges by injecting pure Hawaiian koa extract into your veins while humming "I'm Yours," but this hasn't yet been approved by the FDA. Even after you obtain every uke you think you'll ever need, you'll discover that you're missing a certain size in a certain type of wood, and MGM will just happen to have one for a price you can't refuse.
UAS is still a lot less destructive than GAS. My dad suffers from GAS despite never actually playing the guitar, so our house is filled with big, heavy cases that block access to the upstairs bathroom and take up valuable floor space in almost every room in our small house. I'd gladly trade a guitar for a uke behind his back; even if I got four more ukes, my collection would still fit on my couch with some sitting room left over.