Amazingly, the canonical section of $\mathcal{O}_X(D)$ is the constant 1: $$s_D=1 \in\Gamma(X,\mathcal{O}(D))\subset K(X)$$
Indeed we have isomorphisms of sheaves (=trivializations) of $\mathcal O_{U_\alpha}$-modules $$g _\alpha:\mathcal{O}_X(D)|U_\alpha=\frac {1}{\eta_\alpha}\mathcal O_{U_\alpha}\stackrel {\cong}{\to} \mathcal O_{U_\alpha}:t\mapsto t\cdot \eta_\alpha$$ and $g_{\alpha\beta}=g_\alpha\circ g_\beta^{-1}=\frac{\eta_\alpha}{\eta_\beta}$ is the transition cocycle associated to the covering $(U_\alpha)$ for the line bundle $\mathcal{O}(D)$.
[my convention is the inverse of yours because usually trivializations go from the restriction of the bundle to the trivial bundle while you go in the opposite direction]
So the section $1|U_\alpha=s_D|U_\alpha \in \Gamma(U_\alpha,\mathcal{O}(D))$ is sent by $g_\alpha$ to $1\cdot \eta_\alpha=\eta_\alpha\in \Gamma(U_\alpha,\mathcal{O})$ and of course we have $\eta_\alpha=g_{\alpha\beta}\cdot\eta_\beta$ on $U_{\alpha\beta}$ by the definition of $g_{\alpha\beta}$.
To sum up, the canonical section $s_D=1 \in\Gamma(X,\mathcal{O}(D))\subset K(X)$ is represented in the given trivializations $g_\alpha$ of $\mathcal O(D)$ over the $U_\alpha$'s by the family of regular functions $\eta_\alpha\in \Gamma(U_\alpha,\mathcal{O}) $
A preliminary comment: you ask
Does such a global section correspond to a meromorphic top form (giving a divisor in the divisor class of $K_X$?
No, a section of $K_X^{-1}$ would give a divisor in the class $-|K_X|$. In general, for any line bundle $L$ with associated divisor class $|L|$, a nonzero global section of $L$ gives a divisor in $|L|$. More pragmatically, why would a section of the dual of $K_X$ have anything to do with meromorphic sections of $K_X$ itself?
Now to your (edited) question: here the answer is again no. Let's give some counterexamples.
Example 1: Let $X$ be an Enriques surface. This is a particular kind of minimal surface with the property that $K_X \ncong O_X$ but $K_X^2 \cong O_X$. These two facts already say that $K_X$ cannot have any nonzero global sections, because the square of such a section would give a nonconstant section of $O_X$. Moreover, $K_X^2 \cong O_X$ means that $K_X^{-1} \cong K_X$, so $H^0(X,K_X^{-1}) = 0$ too.
Example 2: If that example seems a little tricky, here's one to convince you that this behaviour is not something special, but is more like the general case. Let $X$ be the blowup of $\mathbf P^2$ in $r \geq 10$ points which do not lie on a cubic. Certainly $K_X$ has no sections, since $\operatorname{dim} H^0(X,K_X)$ is a birational invariant, and on $\mathbf P^2$ there are no such sections. What about $K_X^{-1}$? Well, the associated divisor class is $-|K_X|=3H-E_1-\cdots-E_r$, where $H$ is the class of a line on $\mathbf P^2$ (pulled back to $X$) and $E_i$ are the classes of the exceptional divisors. An effective divisor in this class would have to be the proper transform on $X$ of a cubic curve on $\mathbf P^2$ passing through the $r$ points, but by assumption no such curve exists.
Closing remark: Here is a general remark about what one should expect in questions like this. Inside the real vector space $N^1(X) := \left(\operatorname{Pic(X)}/\equiv \right) \otimes \mathbf R$ of numerical equivalence classes of line bundles, the points represented by line bundles with nonzero global sections form a convex cone called the effective cone $\operatorname{Eff}(X)$ of $X$. The criteria of Kleiman and Nakai–Moishezon imply that the closure $\overline{\operatorname{Eff}(X)}$ is a strictly convex cone, meaning that it contains no 1-dimensional subspaces. That means that, if the Picard number of $X$ is at least 2, then the union $\overline{\operatorname{Eff}(X)} \cup - \overline{\operatorname{Eff}(X)}$ of this cone with its negative cannot fill up all of $N^1(X)$. So there will be lots of line bundles $L$ such that neither $L$ nor $L^{-1}$ have global sections. Example 2 above shows that, in particular, this can happen for $L=K_X$. (Example 1 is slightly different in nature — here $K_X$ maps to $0 \in N^1(X)$ because it is torsion.)
Best Answer
You can cook up a compatible nonzero global section of $L \otimes_{\mathscr{O}_X} K_X$ because $L \otimes K_X|_{U}$ is a flasque on each open subset $U \subseteq X$ where $L$ is trivialized. Choose any nonzero section over any trivialization, restrict to overlaps with other trivializations, pull it back up to an element that restricts to this restriction (under a different trivialization), then glue.