At first I didn't understand why you would think that a "block universe" implies living your life again. Your life is only there in the block once, why would you experience it twice?
But I remembered one approach to this idea, which says that the experience of time comes from your consciousness moving through the block. The block itself is unchanging, but your consciousness starts at the first moment of your life, then moves to the next moment, and continues all the way to the end of your life.
If you think like that, then yes, your consciousness could then go back to the beginning and play through it all again. But it might also jump to another life, or play through your life backwards.
All of these ideas conflict with the spirit of the Block Universe idea, because the spirit of the idea is that time is just like space, it's just another direction within the (four-dimensional) block. But these ideas with the "moving point of awareness" reintroduce time as something external to the block. The block is unchanging, but the point of awareness changes its location within the block.
The reason that these extra ideas, contrary to the intention of the Block Universe idea, are proposed, is because the Block Universe tries to remove the element of change from time; but subjectively we know that time does involve change. So change, e.g. in the form of a moving point of awareness, is reintroduced as something independent of the Block - which is meant to be the whole of reality. Or, people say that change is an illusion: there is no actual flow of time, there are only independent static moments of awareness, each of which has an illusion of flowing into another moment.
Years ago I posted my own compromise answer to these contradictions. The relevance to your own problem is that, I reject the dualism according to which the Block is an unchanging thing and my awareness is something separate from it - an assumption which I consider necessary for the kind of scenarios that you entertain. Change is real; time is passing, the Block is "flowing" at every point and in every moment; and my experience is something like the flow occurring along the worldline of the conscious part of my brain.
The motivation within physics for a Block Universe is that in relativity, the absoluteness of the distinction between time and space is reduced. Instead of space being something which changes everywhere all at once in each moment, the order in time of distant events does not seem to be an objective thing. If, say, two stars blow up at almost the same time, but on opposite sides of the galaxy, there seem to be "reference frames" that are equally valid, but which would disagree about which explosion happened first. And physicists now think that in such cases, there is no objective answer to "which blew up first?" What is objective is the space-time description, and then you can put on a time coordinate in different ways, just like you can put coordinates on a map in different ways.
You can see the details of all that in any encyclopedic article on special relativity. But the question then is, how do we relate this spatialized concept of time, to the idea of time as the medium of change, that we know from conscious experience? Many physicists have decided that this subjective time is somehow unreal, an illusion, something for neuroscience to account for.
There can be psychological motivations for this denial of time-as-change. The pain of change and loss may seem less total, if you can think of the vanished past as still existing in eternity. There is also the happiness of comprehensibility. Earlier I mentioned the difficulty of thinking of a four-dimensional "block" which nonetheless "flows". It is easier to think of something static. So if time is somehow just another direction in four-dimensional space, the metaphysical challenge of understanding what change is, especially in the universe of relativity, can be avoided.
However, it is not actually essential to think about physics in this way. In the next stage of relativity theory, general relativity, you no longer just have relativity of time and simultaneity, you have gravity as arising from "curvature" of space. I should say, by the way, that although relativity doesn't have universal objective time, it has objective notions of spacelike and timelike directions. If you move faster than light inside the block, you would be going in a spacelike direction; if you move within the bounds of light (the only sort of motion thought to be actually possible), you move in a timelike direction. And in general relativity, for a particular timelike trajectory, there is an objective notion of time defined along that trajectory, the "proper time".
I believe, I speculate, that this is part of the way to reestablish the notion of a flowing, changing time within the framework of general relativity. Another aspect would be to focus on matter, the worldlines followed by actual particles (as opposed to the purely geometric worldlines), an emphasis which has something in common with the ideas of Ernst Mach. The idea, then, would be that there is a kind of locally flowing time, within each material system; the proper time of that object.
There is already a problem in understanding how conscious experience relates to the world as described by physics. This is sometimes called the hard problem of consciousness. So I'm saying that there is a subproblem of this "hard problem", the problem of how the subjectively experienced flow of time relates to the blocklike description of space-time in physics, and my proposal is that part of the answer is, that the subjective flow of time, is related to the proper time of the conscious part of the brain.
These are just my speculations, though probably other people have had similar ideas. But I would assure you that no-one has any better ideas, and ideas which entirely deny the reality of physical change are not to be believed, as they lead either to a pathological denial of the reality of experienced change, or to dualistic ideas in which the mind that experiences change is separate from the unchanging universe.