Is it possible to be allergic to strawberry leaves but not the strawberry itself?

Possible. It is definitely a possibility although it is usually the fruit that causes the problem. I am sure the oils in the leaves can still cause reactions.
Yes. Some folks develop hives when they touch the leaves of some plants. This is called contact urticaria and is common with juniper, squash, pumpkin and tomato. They don't necessarily have allergy symptoms when they eat the fruit of the plant. I haven't heard of anyone else with strawberry leaf allergy but it's certainly possible. Get someone else to do the picking & enjoy the eating.