Yeah, no, you're going to have to learn to pee on the side of the road. There's an art to finding the right spot. Trees/bushes are pretty obvious, but culverts are good too. Small variations in terrain can be enough. Start looking for a spot before the need is urgent. Review the terrain ahead, and if you're about to leave the last privacy for a long time, avail yourself of it. Watch where you step. Learn the poisonous plants in the area you'll be traveling.
If all else fails, there's always the ostrich principle - if you can't see them, they can't see you - close your eyes.
And it's not OK to leave TP. If you use it, bring it with you or burn it if it's safe to do so.
And yeah, the rain jacket with the long tail? That does have additional uses. But they are usually bright colored, so that can be a problem.
Icky as this may sound to you, it won't take long to get used to it, please please do not let this prevent you from doing your trip!