Your structure is an endless dense linear ordering, and this is a theory that admits elimination of quantifiers. Thus, every assertion in this language is equivalent to a quantifier-free assertion. If one has finitely many parameters, then the only possible consistent types are the assertions about how those parameters are ordered, and how the new variable $x$ fits into the resulting intervals. That is, the variable $x$ is either equal to one of them, or between two successive ones, or above all or below all of them. All such types are already realized in $\mathbb{R}$, since it has such points already for each of these possible patterns, and so the structure is $\omega$-saturated.