First of all, try pads instead of loo roll. Your dad should not be embarrassed if you toss some in the cart when shopping. Maxi pads can hold a lot, so can mini-pads.
You should talk to your doc about the incontinence, but just tell your dad that you want a checkup. He might think it's about your periods, that's fine. He does not have to be in the room, you have the right to privacy in a medical checkup. See if you can get a woman doctor. In the meantime -- pee about once an hour. It will help your bladder get back into shape.