I'm not sure what the answer is, but it can't be what you suggest. Your conditions allow $a=3$, $b=6$,  $d=7$. But ${\bf Q}(\alpha+\beta)$ is contained in $K={\bf Q}(\alpha,\beta)$, so $d$ must divide the degree of $K$, which must be a multiple of 6 (since it contains ${\bf Q}(\beta)$) but no greater than 18.

I'm not sure what the answer is, but it can't be what you suggest. Your conditions allow $a=3$, $b=6$,  $d=7$. But ${\bf Q}(\alpha+\beta)$ is contained in $K={\bf Q}(\alpha,\beta)$, so $d$ must divide the degree of $K$, which must be a multiple of 6 (since it contains ${\bf Q}(\beta)$) but no greater than 18.