Two things. For one, the striping is most likely a VRAM conflict, although it's been a while since I did DS dev last. Double check where you're copying your graphics and your tile data. Mode 5 uses two text backgrounds, which should be behaving exactly the same if they're initialized a certain way, so I'm not sure what's going on there.
In any case, since you have two backgrounds working, why not just set their priorities in their registers to swap their z order? Background are only drawn in a particular order by default, you can set their priority to have the system draw them in any order you like.