I've never seen the term 'normal' being used in the way of the OP. A semigroup $S$ with the property that $xS = Sx$ for all $x \in S$ is commonly called either duo or normalizing (see also [this answer][1]). Cancellative, non-commutative examples abound. 

For instance, the multiplicative monoid $R^\bullet$ of the non-zero elements of a left (or right) discrete valutation domain $R$ is cancellative, duo, and strongly Archimedean (the latter means that, for every $a \in R^\bullet$, there exists an integer $n \ge 1$ such that any product of any $n$ non-zero non-units of $R$ is divisible by $a$ in $R^\bullet$). The example with skew power series mentioned by user46855 in [their answer][2] is a special case, see Exercise 19.7 in the 2003 edition of Lam's *Exercises in Classical Ring Theory*.


  [1]: https://mathoverflow.net/questions/415718/a-name-for-semigroups-in-which-left-and-right-principal-ideals-coincide/416640#416640
  [2]: https://mathoverflow.net/a/160741/16537