No. Suppose $a$ is a random real over $L$. Then $a$ is random over any $L_\alpha$ satisfying $ZF^-$ plus "$\mathbb R$ exists." If there were $x$ as in your question, then it would be Add$(\omega,\alpha)$-generic over $L_\beta$. But by the well-known orthogonality of random and Cohen forcing, $L[x]$ thinks there are no random reals over $L$, and $L[a]$ thinks there are no Cohen reals over $L$.

We could also do this assuming $\alpha$ is an $L$-cardinal which is countable in $L[a]$, but the above argument also covers many $\alpha$ which are countable in $L$.

No. Suppose $a$ is a random real over $L$. Then $a$ is random over any $L_\alpha$ satisfying $ZF^-$ plus "$\mathbb R$ exists." If there were $x$ as in your question, then it would be $Add(\omega,\alpha)$-generic over $L_\beta$. But by the well-known orthogonality of random and Cohen forcing, $L[x]$ thinks there are no random reals over $L$, and $L[a]$ thinks there are no Cohen reals over $L$.

We could also do this assuming $\alpha$ is an $L$-cardinal which is countable in $L[a]$, but the above argument also covers many $\alpha$ which are countable in $L$.