Key construction in the Full Cramer-Shoup cryptosystem
Yes, if you allow to introduce other primitives into the PKE scheme.
The Kurosawa-Desmedt scheme is an example whose secret key consists of four exponents in $\mathbb{Z}_q$.

Let us fix a group $\mathbb{G}$ of prime order $q$. Let $H$ be a secure hash function (TCR security). 

Cramer and Shoup

Let us review the Cramer-Shoup scheme. As you wrote, the scheme is summarized as follows:


$\mathrm{pk} = (c,d,h) = (g_1^{x_1} g_2^{x_2}, g_1^{y_1} g_2^{y_2},g_1^{z_1} g_2^{z_2}) \in \mathbb{G}^3$
$\mathrm{sk} = (x_1,x_2,y_1,y_2,z_1,z_2) \in \mathbb{Z}_q^6$
$\mathrm{ct} = (u_1, u_2, e, v)$,
where $u_1 = g_1^r$, $u_2 = g_2^r$, $K = h^r$, 
$e = E_K(m)$, $\alpha = H(u_1,u_2,e) \in \mathbb{Z}_q$,
and $v = c^r d^{r\alpha}$.


I omit the decryption algorithm. The scheme is IND-CCA2 secure if the DDH assumption holds and the SKE scheme ($E_K$) is one-time secure.

Kurosawa and Desmedt (CRYPTO 2004)

Kurosawa and Desmedt proposed a new PKE scheme with more compact secret key, as you wanted.
The scheme is summarized as follows:


$\mathrm{pk} = (c,d) = (g_1^{x_1} g_2^{x_2}, g_1^{y_1} g_2^{y_2}) \in \mathbb{G}^2$
$\mathrm{sk} = (x_1,x_2,y_1,y_2) \in \mathbb{Z}_q^4$
$\mathrm{CT} = (u_1, u_2, e, t)$,
where $u_1 = g_1^r$, $u_2 = g_2^r$, $\alpha = H(u_1, u_2) \in \mathbb{Z}_q$,
$v = c^r d^{r\alpha}$, $(k, K) = KDF(v)$, $e = E_K(m)$, and $t = MAC_k(e)$.


The scheme is IND-CCA2 secure if the DDH assumption holds, the SKE scheme ($E_K$) is one-time secure, and the MAC scheme ($MAC_k$) is secure.
In this scheme, $v$ is used to extract keys for SKE and MAC,
 while in the CS scheme, $v$ is used to serve integrity of $u_1,u_2,e$.
By this change, they successfully removed $h$ from the public key and $z_1,z_2$ from the secret key.

Others

There are tons of PKE/KEM schemes improving the Cramer--Shoup PKE/KEM scheme;
efficient ones based on slightly stronger assumptions, say, GDH, HDH, and GHDH assumptions, and less efficient ones based on weaker assumption. Some researchers try to make schemes compact by employing the random oracles. You can find them by search engines, say, Google Scholar.