Let $k$ be a perfect field. I am looking for a $k$-algebra $R$ with the following properties.

$R$ is of finite type over $k$ and is a domain;

for all ${\mathfrak p}\in{\rm Spec}(R)$, the local ring $R_{\mathfrak p}$ is Cohen-Macaulay and ${\rm emb.\, dim.}(R_{\mathfrak p})-{\rm dim}(R_{\mathfrak p})\leq1$;

for at least one ${\mathfrak p}\in{\rm Spec}(R)$, the local ring $R_{\mathfrak p}$ is not Gorenstein.

I would also be happy with the weaker condition "not a complete intersection" in place of "not Gorenstein".

A related question: is there an example of reduced local noetherian ring $T$, which is Cohen-Macaulay, has ${\rm emb.\, dim.}(T)-{\rm dim}(T)=1$, but is not Gorenstein (resp. not a complete intersection)?

Any help would be appreciated.