At the county level, your best bet is to use the county name. You can also geocode the locations, but this takes away the ability to use a filled map for the counties, which may be something you are interested. You are correct in asserting the ambiguity of county names, but they are only ambiguous when not tied to a state.
As an example, I have chosen what a quick Google search revealed to be the most common county name: Washington County, at a grand total of 31 occurrences. As you said, these are ambiguous as a whole, but not when taking state into account, as there is only 1 county of the same name in any given state. You only need to specify that Tableau needs to look at the state field to assign the locations. You can reach this by either selecting the message that pops up in the lower right of your map when you have ambiguous locations (looks something like the following),
or you can go to Map >> Edit Locations and tell Tableau to pull the state from whatever your state dimension is. See the attached workbook for an example.
County Example.twbx.zip 39.2 KB