Diamondstone, Greenberg and I showed that there is a structure with spectrum precisely the array non-computable degrees. Since this contains a minimal pair, its cospectrum is just the computable reals. Every A.N.C. computes an escaping function, but they do not all compute dominating functions, so this structure is self-escaping but not self-dominating.

Edit: Prior to that, Csima and Kalimullin showed that there's a structure with spectrum precisely the hyperimmune degrees, which will also do.

Diamondstone, Greenberg and I showed that there is a structure with spectrum precisely the array non-computable degrees. Since this contains a minimal pair, its cospectrum is just the computable reals. Every A.N.C. computes an escaping function, but they do not all compute dominating functions, so this structure is self-escaping but not self-dominating.

Edit: Prior to that, Csima and Kalimullin showed that there's a structure with spectrum precisely the hyperimmune degrees, which will also do.