I'm interested to see you show that determinism vs. randomness is a false dilemma. Not that it isn't - it's just that you haven't explained HOW it's a false dilemma. You just skate right onto the issue "more important for you".
Along the same lines, how is determinism vs. randomness any less of a false dilemma? You say "if my choices are not deterministic, they must be random." But the concept of free-will is another (well established) alternative to determinism.
We have already entered into an old metaphysical forest here, so why not go ahead and think about what exactly randomness is? From the way you describe the "more important" dilemma, randomness is a sort of determinism - when viewed at a systemic level. That is, we often speak of randomness in the terms of probability, as though the mathematical probabilities of a phenomenon are the a priori governing laws that determine that phenomenon's taking place.
An alternative way of seeing random probability is that the laws governing a system are simply unknown to us - though they are determined. We can calculate a-posteriori probabilities of a phenomenon, but it is not the probability itself that governs the system. This is the crucial point.
For example, assume that humans have free will. My favorite sandwich shop serves Reubens on average once every 18 days. But let's further say that the shop owner has some idea to sell Reubens 20 times a year.
Whether the universe is truly governed by random probabilities, or whether the real will (shop owner's) governing the system is just a black box to me, the calculated probabilities look the same.
So I suggest perhaps that yours is also a false dilemma. There seems to me no necessary implication of non-determinism and randomness, because I view randomness from an epistemologically humble position, i.e., that it is an a posteriori description of our lack of understanding.
Maybe in so much as randomness solves the false dilemma of free-will vs. determinism, free-will solves the false dilemma of randomness vs. determinism. A trilemma, perhaps?