RW Metropolis and ARMS fail

Question

I've been trying to estimate a series of simulated Gamma-distributed random variables and its structural parameters with MCMC for a stochastic volatility model. However, both the random walk Metropolis algorithm and the ARMS algorithm (by Gilks et al. (1995)) fail to converge.

In doing so, I monitor acceptance rates between 30-40% for the RW Metropolis. Nevertheless, both algorithms do not converge after 50,000 iterations (with 30,000 being burn-in). Essentially, the distribution of the Gamma series reads as follows $G_{t} \sim \Gamma ( \frac{\Delta}{\nu}, \nu )$ with $\Delta = 1$. I simulated the series with $\nu = 3.0$ but my estimation always returns values $\nu \in [0.8;1.2]$.

Do you have any ideas how to tune the algorithm(s) further to eventually achieve convergence? It is a bit complicated to explain the entire setting, because it is included in a state-space model of a stochastic volatility model with variance-gamma jumps that yields non-standard posterior densities.

EDIT: The state-space model and the parameter distributions read like:
$Y_{t+1} = Y_{t} + \mu\Delta + \sqrt{v_{t}\Delta}\varepsilon^y_{t+1} + J_{t+1} \\ v_{t+1} = v_t + \kappa(\theta - v_t)\Delta + \sigma_v \sqrt{v_t\Delta}\varepsilon^v_{t+1} \\ corr(\varepsilon^y_{t+1},\varepsilon^v_{t+1}) = \rho \\ J_{t+1} = \gamma G_{t+1} + \sigma \sqrt{G_{t+1}} \\ G_{t+1} \sim \Gamma ( \frac{\Delta}{\nu},\nu)$

The proposal distribution for the RW Metropolis is a $t$-Distribution with 6 dof.