I don't know R, but I will make some comments. If you are using ARIMAX than you pre-whiten both series first (create a PDQ ARIMA model) which is what I assume you did. ADF has serious power issues. So you should test for Stationarity not just with it but with one of the test of Stationarity that has that has the opposite null. If both test show there is a trend (non-Stationarity) than your more confident of your results. Logging does not deal with non-Stationarity as far as I know. It deals with variance issues.

I am not sure what you mean by dynamic regression. Different authors use that for different things.