Cohesive sets with degree below some non-high 1-generic degrees?

Terminology:

Cohesive sets: $A\subset \omega$, for each recursively enumerable set $W_e$, either $A\cap W_e$ is finite or $A\cap(\omega\setminus W_e)$ is finite.

Non-high degrees: Degree $a$ such that $a'\not\geq 0''$.

I'm wondering if it is possible to construct a cohesive set using some non-high 1-generic degree as an oracle? i.e. are there $A$ cohesive, and $B$ non-high 1-generic such that $A\leq_T B$? Thanks in advance!

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Take $A$ and $B$ such that $A \le_T B$, $A$ is Cohesive and $B' \not\ge_T 0''$. This implies that $A' \not \ge_T 0''$ and so by the paper mentioned above $A$ computes a diagonally-not-computable function. But no 1-generic can compute such a function hence $B$ cannot be 1-generic.
No but this is not so difficult. Fix a functional $\Phi$, define $W$ by adding any string $\sigma$ such that for some $n$, $\varphi_n(n)\downarrow$ and $\Phi^\sigma(n) \ne \varphi_n(n)$. If $G$ is 1-generic and $G$ meets $W$ the $\Phi^G$ is not DNC, if $G$ avoids $W$ (say at $\sigma$) then $\Phi^G$ cannot be total (otherwise build a computable DNC by looking at convergences compatible with $\sigma$). –  Adam Day Feb 10 '13 at 16:27
@Adam: Thanks for the note. But I believe in your definition of $W$ it should be $\Phi^\sigma(n)=\varphi_n(n)$ –  Jing Zhang Feb 11 '13 at 8:26