I'm looking for an example of a non-semi-local, non-Jacobson domain $A$, having $\dim(A)=1$.
A commutative ring $A$ is non-Jacobson if it has a prime ideal that is not an intersection of maximal ideals. For a one-dimensional domain, that comes down to the zero ideal $0$ being strictly contained in the Jacobson radical $rad(A)$, the intersection of all maximal ideals of $A$.
Such an $A$ is necessarily non-Noetherian, as is shown here: https://math.stackexchange.com/q/840896
Take for example $A$ to be the integral closure of $\mathbb{Z}_{(p)}$ in $\overline{\mathbb{Q}}$. Then $A$ is of dimension $1$ (because it is integral over something of dimension $1$) and a domain. Further, there are infinitely many prime ideals lying over $(p)$ (because there are algebraic field extensions over $\mathbb{Q}$ of arbitrary degree which are unramified over $(p)$), but every maximal ideal of $A$ contains $p$.
If you are more of a geometric minded person - like me - then, of course, the analogous example "take $A$ to be the integral closure of $k[X]_{(X-a)}$ in $\overline{k(X)}$" also works.