You should consult with an attorney for further information about the merits of your claim. You have two years from the date of this incident to file a lawsuit if there is merit to your claim.
If drugs were found in the passenger compartment of a car in which you were traveling, you are lucky that you are not on trial for joint constructive possession of the drugs. It does not matter that the driver was willing to "take the blame" for the drugs. The law prohibits possession, not ownership. Possession includes more than when it is under your direct physical control, such as when its in your pocket. Possession can be joint (by two or more people) and includes constructive possession (where the item is in an area where you could gain physical control of it and the State can prove circumstantially that you had the intent to gain control of it in the future. Therefore, when you are travelling in the same car as a quantity of dope, the police can arrest you and the State can circumstantially prove that you possessed it. You do not have a claim for false arrest.
Be very happy that they dropped all the charges against you -- very happy indeed. Having said that, you still have an arrest record for the drug charges. Hire legal counsel to work on an expungement of that record. As to any lawsuit against the Douglas County SO, you would have little chance of prevailing based on the limited facts you have presented. Good luck.