Are you on formal probation? It sounds like you have just one DUI and are on probation for that. Typically a term of misdemeanor probation does not allow a police officer to search your house. If you are on felony probation that can be different. If you are on informal probation you are free to leave the state; just make sure you stay current on your court payments and have completed your DUI classes. If you are on felony probation you need to get permission from the court.
Typically, a DUI results in a misdemeanor conviction on your record and informal summary probation. If you are on felony/formal probation, the process of transferring to Washington will take time and is contingent upon your probation officer approving it. However, if you are not on formal probation, there should be nothing preventing you from leaving the state. It is strange for an officer to come and conduct a search of a probationer on informal probation. There is no way that I'm aware of to transfer the fine to Washington. It will still be due and payable to the court in which you were convicted.
You are not required to live in the state of your conviction if you are on informal probation. However, if you did not pay your fines or do classes, there may be a warrant for your arrest. That is probably why there were police at your door. Contact an attorney to assist you in recalling the warrants, help finding classes in another state, and setting up a payment plan with the court.
If you had a lawyer for this case, check with them for your conditions of probation. You can also get the conditions from the clerk's office for the court where you were convicted. Moving out of state should not be a problem if you are on unsupervised probation, as there is nothing to transfer. However, usually you are required to let the court know when you change your address while you are still on unsupervised probation. If this is a federal case, you do have to check with your probation officer and ask permission to move. That would be more difficult.