Even if there is something that could influence our choice but that is not the state of the universe and that is not randomness, it would still be something, not yourself, pulling your strings, would it not?

It could be, for example if it was a god that influenced your choices and assuming we would neither consider this god part of the universe nor part of you. But that is also not necessarily the case, for example if your choices would be influenced by your soul assuming we would not consider souls part of the universe but still part of you.

But it is also not obvious that this is possible in a meaningful way. Whether souls or gods, they must at least be able to affect you and your choices, otherwise they may also not exist at all, at least if we are only thinking about your lifetime in this universe, they could still of have consequences in a life after death or whatever. One could also argue that you, meaning the part of you within the universe, must be able to influence your soul because in which sense would it be your soul if it were unaffected by everything you experience and do? And similarly we usually think of a god as being influenced by us by observing what is going on in the universe.

So we have at least a one-way interaction between us and the universe in general and souls or gods. And for many interesting things we want to get out of souls or gods we would even need two-way interactions. Which raises the question why we would not consider them parts of the universe? And this is the case for all supernatural things, why would we not consider them part of the universe, after all they must at least be able to influence us at least by being observable. We could exclude them, for example ghosts, because we have no understanding of how they work, but that is a rather weak criterion not unlike yet undiscovered physics.

Or we could try to exclude them because the usual laws of nature do not apply to them [1]. But in order to be able to move the curtains in the bed room or to tilt all the picture frames in the living room, a ghost must at least be able to interact with the stuff in our universe, otherwise it would remain unobservable, and therefore supernatural things can not be totally disconnected from the normal universe and how it works. So it is not obvious to me that you even can separate things into a normal universe and another realm of souls, gods, or ghosts that is however still interacting with the normal universe without this distinction being rather arbitrary. Especially if you think of the universe as all the things that can potentially interact with each other, possibly only in a one-way fashion [2], than the universe is closed under this and everything outside the universe can not possibly affect anything in the universe by definition.

To return to the original topic, it may or may not be you that pulls the strings if your choices are influenced by something outside of the universe, but I think the hard part is actually finding something outside the universe that could have an influence on your choices and that is outside of the universe in a meaningful way and not just due to an arbitrary definition like well understood things are in the universe and things we do not yet understand are outside of it.

[1] Just as in the case of the universe with its laws of nature and free will one can ask what kind of rules can, could, or have to apply to souls and gods. What can affect their decisions, what can be affected by their decisions? Is a god or soul subject to any laws of nature or at least logic or whatever? And if not, what determines its behavior, it probably should not be randomness. Note that this are very similar questions as compared to free will, just as if we only pushed the hard parts of the question out of the universe but that of course does not answer them. Even if we conclude that our choices are influenced by our soul, that just means we now have to figure out how the soul works and makes decisions leaving us essentially with the same questions as before.

[2] Radioactive decay is an example, the decay certainly affects the universe but you can not do anything to affect the decay of a nucleus. At least in the classical picture of radioactivity, the quantum Zeno effect actually allows influencing decay rates.