You can find a measure of introversion and many personality measures at the International Personality Item Pool (IPIP) website: http://ipip.ori.org/ipip/. The most popular social desirability scale is the Crowne-Marlowe. There is a short form, but I don't have the cite.
Crowne, D. P., & Marlowe, D. (1960). A new scale of social desirability independent of psychopathology. Journal of Consulting Psychology, 24, 349-354.
Crowne, D. P., & Marlowe, D. (1964). The approval motive. New York: John Wiley.
The standard Eysenck/Maudsley personality inventories, which you would probably use to measure introversion, also have a 'lie scale' which effectively measures the tendency to give socially acceptable answers.