So I will try to make a leap in logic to what I think you must believe and you correct me if I'm wrong. Our actions/decisions determine which "world" our consciousness flows into, perhaps?
If so, is there a separate consciousness that is active in each world?
Consciousness is irrelevant. This is fundamentally physical.
I think it is easiest to try and visualize it, then work backward to conceptualization.
This one is a very basic model of what I believe ultimate reality would entail. The middle would be the present. The left would be the past. The right would be the future. Each "arm" on the left or right is a different phase space. As you can see, there are nine (3 x 3) different phase spaces which would correctly incorporate the current slice of space-time, otherwise known as the present. Each phase space can have their own natural laws.
As you can see, there are many different possibilities of the past producing the present and many different possibilities of the future from the present. Our actions in the present can change the course of a phase space through our slice (the present) to a future slice of space-time.
Oh, and under Eternalism, all these different phase spaces exists simultaneously.
So with this model in mind, you can see how the future is indeterminate based upon the present. However, if one was to just look at an individual phase space as a whole (from the beginning of time to the end), then the implications would be different. Imagine a straw (the entire phase space). If you are at the end of the straw, looking back though time towards the beginning, it would seem as though our actions were determined. In fact, they are within a phase space. The interaction between a particular phase space and a slice of time is the only thing keeping us from being completely determined outcomes. I guess such a complex interaction is more commonly dubbed as "free will".
Here is another picture to help:
Each dot represents a slice of space-time. Under MWI, the split would be into a different space-time slice. The connection of these splices through time is what makes up a particular phase space. You can imagine a line connecting the dots in an array of different ways.
I hope this helps.