Kaylag, you're doing everything right. Just express a little into a towel and then let baby nurse. This happened with me and both of my children. My first baby would nurse until I let down and then pop off and cry while milk sprayed everywhere then she'd latch back on and nurse peacefully to sleep. Then I learned to wait until I had let down the first time to nurse (I started to leak as soon as baby showed interest, or looked at me, or smiled, or I looked at picture, etc). As she grew, she would just hold on and swallow fast until we got regulated and then around 8 weeks the problen fixed itself.
My concern is if you begin pumping so soon then you will have an oversupply issue. There isn't any reason you can't wait and begin pumping the milk left in your breast when baby is finished nursing if you need a freezer store for when you leave.
Hope this helos.