My first advice, as I do not practice in either state, is to contact a lawyer in the state where you now live. Perhaps it would be as simple as filing an action in Court to declare your marriage. This of course depends on you being able to prove that you satisfied any statutory obligations in Iowa for common law marriage. Again, though, you will likely need to speak to an Iowa attorney because you need to verify whether Iowa still recognizes common law marriages.
Minnesota does not recognize common law marriage. Minnesota considers you to be cohabitants, not husband and wife. The simplest way to insure that Minnesota will recognize your marriage is to get married formally. Other than that, you should talk with an attorney to see how best to arrange your legal affairs so that they approximate the legal status provided by a formal marriage.