I wouldn't say it is impossible to get something from DATY...unlikely, improbable, but not impossible.

That said...a few hours afterwards would be pretty far fetched for it to include chills and a fever.

Sore throat could be irritation from a soap, parfume, lotion, douch, lube, latex, etc and it could show up in a relatively short amount of time, but for an infection to permeate your system, to the extent that you had sympoms including fever and chills...not likely for most common and most uncommon infections to do that.

On avaerage it takes aproximately three days to a week from exposure, for a comon cold or flu become symptomatic.  For STIs, it can take days to months depending on the infection.

So, I highly doubt any issues with your throat other than either a mild allergic reaction to some chemical, like soap, perfume, etc...or perhaps even that you tend not to have your mouth open and tongue out like that on a daily basis and your throat dries some causing irritation.  Chances of it being any sort of infection that would be that quick to infect your system to be symtomatic in a matter of few hours would probably not be anything that would clear up on its own soon after.

But that just a guess really.