Existence results for primitive elements in cubic and quartic extensions of a finite field

With $\text{Fq}$ the finite field of $q$ elements, we investigate the following question. If $\gamma$ generates $\text{Fqn}$ over $\text{Fq}$ and $\beta$ is a non-zero element of $\text{Fqn}$, is there always an $a\in \text{Fq}$ such that $\beta (\gamma +a)$ is a primitive element? We resolve this case when $n=3$, thereby proving a conjecture by Cohen. We also improve substantially on what is known when $n=4$.