I hate to say this, but if you are going to show western pleasure you are required to use split reins. At most shows, if you use rope reins you will be DQed. While practicing, you can tie the leather split reins together at the ends, but I would really practice getting your grip on the reins more secure so you will be comfortable with them. If it requires starting with nylon split reins then moving to leather split reins, then do that. But for WP, you have to use split reins. Also, if you are in a curb bit you have to show one handed. If your filly is 5yo or under, then you can show in a snaffle or bosel and use two hands. For the bosel, the reins are not split, for the snaffle, they still are. I can tell you are nervous about it, but you do need to work up to using them if you plan on showing WP (or horsemanship, reining, trail, etc). Best of luck, and just take small, easy steps to get yourself comfortable.
Oh, and I have never held onto the reins when I have been thrown. Just doesn't happen with me, and in all honesty, most horses won't go that far.