Busan to Seoul via bicycle

I am looking to bicycle to Seoul from Busan over Chuseok but am wondering about the condition of the bike paths on the 4-Major River Project that the previous president initiated. Specifically, are any of them paved? Most of them? I'm asking because I only have a road bike (8-speed, thin tires, aka. NOT a mountain/trail bike). Anyone know enough to say whether I'd be able to tackle it or do I have to look at getting another bike. If it's the latter I'm not going to do and will just have to wait until a more opportune time arises. Anyway, any advice much obliged. Thanks.