Before stating my question, let me recall (part of) the classical result on the adjunction map for complex projective surfaces, due in this modern form to Beltrametti and Sommese:
Adjunction Theorem. Let $X \subset \mathbb{P}^n$ be smooth surface and $D$ the hyperplane class. If $(K_X+D)^2 >0$ then (apart from a finite list of exceptional cases) the adjunction map $$\varphi_{|K_X+D|} \colon X \longrightarrow X_1 \subset \mathbb{P}^N$$ defined by the complete linear system $|K_X +D|$ is birational onto a smooth surface $X_1$ and blows down precisely the $(-1)$-curves $E$ on $X$ with $DE=1.$
Now, I have a rational surface $X$ with an effective, ample and basepoint-free (but not very ample) divisor $D$ such that $h^0(X, \, D)=3$ and $D^2=3$, inducing a triple cover $f \colon X \longrightarrow \mathbb{P}^2$.
I have many computational evidences showing that the first adjunction map associated with the complete linear system $|K_X + D|$ should be birational and contract precisely the $(-1)$-curves $E$ on $X$ such that $DE=1$. Unfortunately, in order to rigorously prove this I cannot use the Adjunction Theorem in the form stated above, since $D$ is not very ample. So my question is:
Question. Is there any version of the Adjunction Theorem that holds for $D$ ample (maybe with some additional assumption) and that I can use in my situation?
Any answer, suggestion or reference to the existing literature will be greatly appreciated.
Edit. Let me add the description of one of the simplest examples I have (the others are more complicated, and I'm not yet able to understand them so explicitly as this one).
Let $X$ be the blow up of $\mathbb{P}^2$ at $13$ points that impose only $12$ independent conditions to plane quartics (we can construct such a set of points by a Cayley-Bacharach argument). Let $L$ be the strict transform of a line in $X$ and $E_i$ the excepitional divisors. Setting $$D = 4L - \sum_{i=1}^{13}E_i$$ we have $$D^2=16-13=3, \quad h^0(X, \, D)=15-12=3.$$ Notice that $K_X+D=L$, hence $h^0(X, \, K_X+D)=3$ and the adjunction map $\varphi_{|K_X+D|}$ is birational in this case, in fact it is precisely the blow-down morphism $X \longrightarrow \mathbb{P}^2$.