Chlamydia rarely infects the throat and is rarely transmitted by oral sex, and never known to be transmitted by kissing
. Gonorrhea can be in the throat
, but not known to be transmitted by kissing
; and usually doesn't cause sore throat. (The vast majority of oral gonorrhea infections are entirely asymptomatic.) And no, gonorrhea does not travel through the body to show up at additional anatomic sites.
I recommend against STD testing, assuming this is your only potential STD exposure.