The definition of the amalgamated coproduct $A(I)$ of two copies of a partially ordered monoid $S$ is as follows:
Let $I$ be a right ideal of a pomonoid $S$, $x,y,z$ not belonging to $S$, and $$A(I)=(\{ x,y\} \times (S\setminus I))\cup (\{ z\} \times I),$$ and the definition of a right $S$-action on $A(I)$ is as follows: $$(w,u)s=\begin{cases}(w,us) & \text{if}\; us\not \in I, \; w\in \{ x,y\}\\ (z,us), & \text{if}\; us \in I. \end{cases}$$
My question is: What is $(z,u)s$ if $us\not \in I$?
According to your definition, if $(z, u) \in A(I)$, then $u \in I$. Now since $I$ is an ideal, $us \in I$ for every $s \in S$. Thus the case you consider simply cannot occur.