Your ARIMA model is (probably !) an overfit and (probably ) is the cause of your confusion. What you are looking for might be a transfer function model expressed(a.k.a. ARMAX) expressed as a right-hand side model. https://stats.stackexchange.com/search?q=user%3A3382+transfer+function+model+as+a+right-hand+side
in general ...... where the time lag effects are IDENTIFIED rather than assumed and anomalous data is rendered harmless and trends and level shifts are clarified.