Let $g$ be a cuspidal modular eigenform of weight 2 and level $N$ that has CM, so comes from a Groessencharacter of an imaginary quadratic field $K$. Let $p$ be a prime not dividing $N$.

(1) Is it possible that $g$ can be congruent mod $p$ to an eigenform that does not have CM by $K$?

I suspect that this may be possible [EDIT: it definitely is possible], so I'd also be interested in the following more specific question:

(2) Suppose $g$ is new of level $N$ and "$p$-isolated" (not congruent modulo $p$ to any other eigenform of that weight and level). Let $\ell \ne p$ be a prime not dividing $N$ and split in $K$, and let $g'$ be either of the two eigenforms at level $N\ell$ corresponding to $g$. Can there exist eigenforms of level $N\ell$, congruent to $g'$ mod $p$, that do not have CM by $K$?