The mh_mode file is the mode after the MCMC run, not after mode-finding. Using it as the mode_file and them running mode-finding again (potentially with a different mode-finder) should increase your chances of finding a better mode and getting quicker convergence. If your posterior mode stays constant, your mode-finder most probably got stuck. Otherwise, the issue is not about convergence of the chains to the same mode, but about mixing. In that case, you might want to try using a user-defined jumping covariance matrix.
Regarding, identification: see [Identification problem)