You may be toeing horse. You may also have been moving right off when got in saddle, instead of making horse stand, and may be reinforcing that by the way you halter and go to leading, which also starts a horse to moving like this.
Reins should be shortened to mount, not excessively, and right rein should be shorter than left, so if horse does move, it is moving under you, not away.
Sliding into saddle, can result in you not being able to get down too, if you need to fast, and I don't know that that is something I would do?
You may also be taking to long to come up in stirrup and settling, or dragging toe across back.
Girth may be pinching skin, if you aren't stretching forelegs out to make sure, or you have not tented saddle pad, so that when you are mounting it is pulling mane at withers or pulling skin.
Horses make me a better person.