Iused to work on the check-out in my parents' grocery store in my teens, well before people were as enlightenad about such matters as they are now, and a lot of ladies and girls were embarassed by me serving them with towels or tampons. To be honest, it meant no more to me than if they were buying tea or coffee - it's just an item that we sold in the shop - and that's the attitude you will find from the cashier in your local shop - if you are really worried, it won;t be hard to find one with a lady serving.
Why not tell your dad that you need some deoderant, shampoo, things like that, and you;d like to choose your own now that you are growing up. Add in the cost of the tampons, and ask him for the money - that saves any embarassment all round.
So sorry you have this problem, it's dreadful to loose your mum, and then have this to deal with.
Love, A xx