It's good to see that you're putting some thought into the directions your rivers are taking, but you seem to be missing a fundamental concept.
Rivers don't fan out from a source; they converge from a source.
Somewhere out on those lower plains there is a lowest point. All tributaries are therefore going to flow towards it as they work their way around higher ground in their path. Along the way, they meet and form the main river. Remember that a river system drains a large area that is at a higher level, whether it be a basin, a plain, or a mountain range and it's the very nature of the flows to converge as they meet each other at lower and lower levels until they become one river and flow into the lowest local point that leads to the sea.
Rivers do not split along the way except for maybe a rock or small island. If they flow into a lake they generally flow out of it somewhere and continue their courses to lower and lower ground. There may be more than one influx of water, but there will almost always be only one outflow of water. At one time there may be multiple outlets, but one of those is going to be the lowest and that one will always have the greatest outflow. Over time, erosion will deepen that exit. Even if it's just an inch, that will create a kind of snowball effect; the more water that flows out, the deeper the outlet becomes...the deeper the outlet becomes, the more water flows out. Eventually that will be the only outlet.