Is the County liable?

Regarding a 2 acre parcel of land, adjoining our house: Taxes were 9+ years delinquent on this land, we paid the delinquent taxes and were told by the county attorney's office to notate our check "full and final payment for clear title..." to the abandoned land. The county never recorded a clear title, however, we did file a non-warranty deed (from husband to wife) to strengthen claim.

Approx. one year after this happened, people who live adjoining this land (on the other side from our house) filed a quit-claim stating they bought the land from previous owners (the ones who didn't pay delinquent taxes and who abandoned land).

If the county would have filed clear title at the time we paid delinquent taxes, the "neighbors" could not have purportedly purchased this land.

Any advice on recourse against the county?

I think you are going to have to file suit against the county and/or your neighbors to resolve this issue. It seems to me that you should have checked that title was in fact transferred to you after your check to pay off taxes was cashed. It's unclear when your neighbors bought the land from your the previous owners so it is possible that their claim to the property is superior to yours. Find a real estate lawyer in your state; you can check this site for some names to get you started.