I would animate keep the stroke animation on the same shape layer. I see you have two.
It looks like you have not included the tying the position of the background shape layer to the position of the text layer. Unless you want them to move independently, you shouldn't do that. All you should have to do is animate the position of the text layer and add a Trim Paths animator to the stroke. You can have more than on stroke on a shape (path) and you can add an additional Trim Paths to every stroke you apply to a shape.
If you use time remapping the entry and exit speed of your animation will change. A better option would be to set protected areas for the in and out transitions. There are several good tutorials on Essential Graphics and Mogrts that explain how to do this.