I have ice cream in my show right now. Two different servings actually. One is a sundae on a dish so that's not a concern. But the other is in a waffle cone and is also on stage for a little over four pages like yours. And we've had no dripping problems.
We leave our ice cream in the freezer until a couple scenes before it needs to enter (off the top of my head I'm not sure of the exact timing) at which point the tub of ice cream gets transferred to the refrigerator, so it becomes just a tiny bit soft and very easy to scoop. So we scoop it into the cone while it's soft, then stick the completed cone in the freezer (standing up inside a glass) until immediately before its entrance.
We went back and forth about putting a paper wrapper on the bottom of the cone, but then decided to go with a folded paper napkin wrapped around it instead. That way if it starts dripping at all there's already a napkin onstage to clean it up with. But we haven't had any issues yet!