By "drop-down" you are meaning "Data Validation List". When yes, then it isn't possible. Easiest workaround will be:
On separate sheet create a table, where 1st column lists all 50 States, and in second column are 2-character codes for those States. Define the 1st column of this table as named range. In your original table, set Data Validation List with defined Name as list source (using Names is only way to use tables in another sheets as Data Validation Source). In another column, use VLOOKUP formula to get 2-character codes from States table, depending on your selection you did using Data Validation.