I have searched arround the web and economic journals trying to find out research related to a macroeconomic model in which the country is fully dollarized, but I haven't had any luck. My main concern is how to deal with exchange rates and the monetary authority. Maybe I just have to forget about them.
You are not likely to find New Keynesian DSGE models where money or currency plays an explicit role. These models mostly abstract from these considerations altogether and monetary policy is simply represented by the interest rate. I understand your situation as I myself have tried to write papers and searched for literature where dollarization plays a role.
If you can, try to circumvent the explicit use of currency in your model. Think about what the currency's role is in your model, e.g. if it is to have different inerest rates in different countries or in different currencies, then simply use different interest rates instead of different currencies.
Another way would be to go outside of the standard DSGE framework and use other international finance, open economy models, as you would find in the Obstfeld/Rogoff book.
If you really want to stick with DSGE, I would recommend looking into open economy DSGE models as well as DSGE models with "Money in the Utility Function".
You can use the following paper as a sort of 'benchmark' for exchange rate treatment for a fully dollarized economy.
In this case, you are not following the Dominant Currency Paradigm completely, but partially. While you will have complete pass-through of the Dollar to home economy, the other currencies can still follow the old method of treatment.
(will update with more references soon!)