amoxicillin with clavulanic acid (Curam) is a common cause of allergic drug
rash, which probably explains the
rash on your
hands and feet. No STD is a possible cause. Syphilis cannot cause rash this soon; skin rash occurs with the secondary stage of syphilis, which doesn't begin until at least 3 months after catching it -- and in any case, Curam would have prevented or cured syphilis.
In our previous discuss 3 days ago, I advised STD testing, including gonorrhea and chlamydia testing.