I am aware of the stochastic singularity problems derived from having fewer shocks than observables. Nevertheless, imagine that I consider the SW 2007 model and I erased the risk premium shock and instead, I consider in another that one of the other shocks instead of being as follows:

a = rho * a(-1) + ea

it is as follows

a = rho* a(-1) + ea + eb

where both ea and eb are IID. Then, dynare allows me to estimate the model but I would have just 6 disturbances but 7 observables.

No, this would still be wrong. For one, the two shocks are observationally equivalent and therefore not identifiable. Dynare should give you an error message in this case. Which version are you using?

Identification depends on the set of observed variables. If, as I suspect, aux1 is not observed then your second version does not change anything (actually you coud write directly eb(-1) in the first equation without introducing the auxiliary variable which will be created behind the scene by Dynare’s preprocessor).

Having a news shocks will indeed change the identification properties due to introducing a different timing. In that case stochastic singularity should not be an issue.