  We introduce a two-variable refinement $\hat{Z}_a(q,t)$ of plumbed 3-manifold
invariants $\hat{Z}_a(q)$, which were previously defined for weakly negative
definite plumbed 3-manifolds. We also provide a number of explicit examples in
which we argue the recovering process to obtain $\hat{Z}_a(q)$ from
$\hat{Z}_a(q,t)$ by taking a limit $ t\rightarrow 1 $. For plumbed 3-manifolds
with two high-valency vertices, we analytically compute the limit by using the
explicit integer solutions of quadratic Diophantine equations in two variables.
Based on numerical computations of the recovered $\hat{Z}_a(q)$ for plumbings
with two high-valency vertices, we propose a conjecture that the recovered
$\hat{Z}_a(q)$, if exists, is an invariant for all tree plumbed 3-manifolds.
Finally, we provide a formula of the $\hat{Z}_a(q,t)$ for the connected sum of
plumbed 3-manifolds in terms of those for the components.
