I also have a 99 LT (new owner) and I've managed to do the same thing. I'm stationed in Guantanamo Bay and keep my bike in Jacksonville, FL. I'm in Jax usually once or twice a week and ride when I can. The last time I was there, my bike had a dead battery, so I jumped it with my jeep. It appears there's a design flaw in the starter relay, which caused itself to stick/weld in the on position......so now, whenever I hook up cables to the bike, the starter immediately turns over. Key in, key out.....doesn't matter, it turns over whenever juice is applied. If I turn the key on, the bike will start, but I can hear the starter still whirring along under all that plastic with the motor running.
There's a thread around here talking about a temp fix for the problem without changing the relay...which is the "correct" fix. It involves removing your dead battery and charging it out of the bike (I'm just going to buy a new one...It's due anyway.), then putting it back in. When the bike tries to turn over, with the ignition on, press and release the starter button and it is supposed to unstick your starter relay. It looks like a decent amount of folks have had luck with this procedure. Next time I'm in Jax, I'm going to give it a try. Right now, I'm having a hard time forking out over 200 bucks for a stinkin relay.