In theory, this is straightforward: the $(K+2)$-dimensional vector $(X,Y,Z)$ is jointly normal, hence there exists some deterministic $x$, $\eta$ and $\zeta$ in $\mathbb R^K$ and a centered $K$-dimensional normal random vector $T$, with variance-covariance matrix $C$, independent of $(Y,Z)$, such that conditionally on $(Y,Z)$, $X$ is distributed as $ U=x+Y\eta+Z\zeta+T. $ Thus, conditionally on $A^2=Y^2+Z^2$, $X$ is distributed as $x+T+V$, where $V$ follows the conditional distribution of $Y\eta+Z\zeta$ conditionally on $A^2$.
First step: To find $(x,\eta,\zeta,C)$, one solves the system $ (1)\ \mathrm E(U)=\mathrm E(X),\quad (2)\ \mathrm E(UY)=\mathrm E(XY),\quad (3)\ \mathrm E(UZ)=\mathrm E(XZ), $ and $ (4)\ \mathrm{var}(Y\eta+Z\zeta)+C=\mathrm{var}(X). $ These four equations are equivalent to $(1)\ x+\mu_Y\eta+\mu_Z\zeta=\mu_X$, $ (2)\ x\mu_Y+\Sigma_{YY}\eta+\Sigma_{YZ}\zeta=\Sigma_{XY},\qquad (3)\ x\mu_Z+\Sigma_{YZ}\eta+\Sigma_{ZZ}\zeta=\Sigma_{XZ}, $ and $ (4)\ \Sigma_{YY}\eta\eta^*+\Sigma_{YZ}(\eta\zeta^*+\zeta\eta^*)+\Sigma_{ZZ}\zeta\zeta^*+C=\Sigma_{XX}, $ which, in the general case, determine uniquely the $K$-dimensional vectors $x$, $\eta$ and $\zeta$ and the $K\times K$ matrix $C$. (Note that $\mu_Y$, $\mu_Z$, $\Sigma_{YY}$, $\Sigma_{YZ}=\Sigma_{ZY}$ and $\Sigma_{ZZ}$ are real numbers while $x$, $\eta$, $\zeta$, $\Sigma_{XY}$ and $\Sigma_{XZ}$ are $K$-dimensional vectors and $\Sigma_{XX}$, $\eta\eta^*$, $\eta\zeta^*$, $\zeta\eta^*$, $\zeta\zeta^*$ and $C$ are $K\times K$ matrices.)
Second step: Basically, a two dimensional gaussian vector $(Y,Z)$ with prescribed distribution is given and one looks for the distribution of the vector combination $Y\eta+Z\zeta$ conditionally on $A^2=Y^2+Z^2$, for possibly any $\eta$ and $\zeta$ in $\mathbb R^K$.
There exists some i.i.d. standard gaussian vector $(Y_0,Z_0)$ and a $2\times2$ matrix $B$ such that $(Y,Z)=M+(Y_0,Z_0)B$, where $M=(\mu_Y,\mu_Z)$ and $B^*B$ is the variance-covariance matrix of $(Y,Z)^*$. Thus, $A^2=a^2$ corresponds to an ellipse in the $(Y_0,Z_0)$ plane and one seeks the distribution of some ($K$-dimensional) affine combination of $(Y_0,Z_0)$ conditionally on the fact that $(Y_0,Z_0)$ is on this ellipse.
I doubt that some convenient formula exists in the general case, but maybe some more specific hypotheses are made.