$A_{sa}$ is the set of self-adjoint elements in $A$.
$GL(A)$ is the set of invertible elements in $A$ and $GL(A_{sa})$ is the set of invertible elements in $A_{sa}$.
I need an example where $GL(A_{sa})$ is dense in $A_{sa}$ while $GL(A)$ isn't dense in $A$, but I can't imagine which kind of $C^*$-algebra has such property.
An example where $GL(A)$ isn't dense in $A$ is $C([0,1]\times [0,1])$. And furthermore, when $X$ is a compact metrix space , $GL(C(X))$ is dense in $C(X)$ iff $\text{dim}(X)\leq 1$.
However, $GL(C(X)_{sa})$ is dense in $C(X)_{sa}$ iff $\text{dim}(X)=0$. So such kind of algebra doesn't have the property.
By the way, $GL(A_{sa})$ is dense in $A_{sa}$ iff every unitary $u\in U_0(A)$ can be approximated by unitary with finite spectrum. Don't know whether this would help. $U_0(A)$ is the group generated by elements like $e^{ia}$ with $a\in A_{sa}$.
The Cuntz algebra $\mathcal O_n$ satisfies the desired conditions.
Firstly, $GL(\mathcal O_n)$ cannot be dense in $\mathcal O_n$ because this only happens in stably finite C$^*$-algebras.
Secondly, by [1, Corollary 4.7] one has that the real rank of $\mathcal O_n$ is zero which means that the self-adjoint invertibles are dense in the set of self-adjoint elements.
[1] Jeong, J. A.; Park, G. H.; Shin, D. Y., Stable rank and real rank of graph (C^ *)-algebras., Pac. J. Math. 200, No. 2, 331-343 (2001). ZBL1056.46050.