Sorry about that, I forgot that you have actual data (I usually use these in latent variable models).
Probably all you need to do is switch to a non-centered parameterization (similar to how you generated the data). That often does dramatically better with sampling in time series models. Here’s some sample code:
If that doesn’t work and you want to switch to a stationary model, then the most straightforward thing is to difference your data to make it stationary:
\Delta y_n = \beta \Delta y_{n-1} - E + \epsilon_n
This is a slightly different model, especially if you don’t difference E. But generally it’s easier to work with stationary time series models.