That is Yamaha's crappy suspension design. Many claim its due to the front suspension settings, and will try to sell you tons of parts like new (expensive) shocks, sway bar, links, and more. Others try to say its the skis. I fixed ALL of my tippy feeling, and the other problems like pushing in the corners, twitchy feeling at high speeds, etc. with one move- replacing the rear suspension. I am surprised your '05 feels tippy, because it has a redesigned rear suspension, but I guess they missed the target on correcting the handling with it.
My wife had tried riding my sled when it was all stock, and she had similar feelings as you...it was very tippy (which scared her), and it rode terrible (your '05 suspension should have fixed that). After I installed an Artic Cat rear suspension, she tried it and I couldn't get my sled back until we got home! It corners very flat, doesn't push in the corners, and it rides very soft and plush, but doesn't bottom out.
The Cat suspension isn't for everyone, and it probably isn't for you since you want to keep your sled stock, but it just shows that the problems are with the way the rear suspension is designed. Otherwise, my handling wouldn't have been improved so much just by replacing the rear suspension.
Jim