I ran into this problem also

Practice! 

If you want to adjust the route it's probably best to add a via point. Press down on the appropriate section of route until the viapoint popup pops up. 

Then, press down on the viapoint and (still pressing down) move it to the desired location. If you're thumb/finger is not in the right place (on the viapoint) or if you lift it briefly and press down again all you'll do is move the map. 

At least that's the Android way.

I'd suggest practicing with a very simple route. Add a via point and practice dragging that around.

By the way, the website on your phone has many advantages over the app for planning, notably an 'undo' function.