Let $X^e \colon \mathbb{N} \to \{\pm 1\}$ be the random, completely multiplicative function whose values at the primes $(X^e(p))_{p \text{ prime}}$ are independent and uniformly distributed on $\{ \pm 1 \}$. Let $X=X^eμ^2$. We determine the order of magnitude of \[ \mathbb{E} \Bigg[ \bigg| \frac{1}{\sqrt{x}}\sum_{n \le x} X^e(n) \bigg|^{2 q} \Bigg]\quad \text{ and }\quad \mathbb{E} \Bigg[ \bigg|\sum_{n \le x} \frac{X(n)}{\sqrt{n}} \bigg|^{2 q} \Bigg] \] for $q\in [0,1]$, showing that a transition from better-than-square-root cancellation to growth like a power of $\log x$ occurs when $q$ crosses $1/2$. In particular, the first absolute moments are of order $(\log\log x)^{3/4}$. The proof proceeds by comparing these sums to integrals of random Euler products, which are then studied by analogy with Gaussian multiplicative chaos integrals. The main difficulty lies in proving the lower bound at $q=1/2$, for which we also adapt arguments from the study of random walks.