$a\geq 0$, $x\sim y$ implies $x+a\sim y+a$ so the preference is linear?
This is not true.

Let $n=1$ and define $u(x)=\min{\{x,0\}}$. Let $\succsim$ be the preference relation represented by $u$. This preference relation is continuous and convex. We also have $x\sim y$ implies $x+a\sim y+a$ for any $a\geq0$ and $x,y\in\mathbb R$. But let $x=0$, $y=1$, and $a=-1$. Then $x\sim y$, but $y+a=0\succ -1=x+a$, thus $x+a\nsim y+a$ and $\succsim$ is not linear.