Thanks for the great suggestions, everyone. They seemed pretty obvious in retrospect, but I'd not thought of them before. I couldn't find conical washers that would accommodate the bolt and fit into the holes in the toilet, so I just got some double-stick tape, and that seems to be working fine.
I actually did some checking and I guess the toilet seat I bought probably did come with some adhesive pads or conical washers, but I must have just torn through the box, grabbed the seat and the bolts, and thrown the other stuff out. Oh well.