Welcome to the STD forum.
Taking treatment without being tested is not the best strategy. But now that you have taken 1.0 g of azithromycin (Z-pak), you don't need any more treatment. As you already know, that dose would cure chlamydia if you had been infected. As for gonorrhea, you really don't need additional treatment at all.
First, gonorrhea of the penis almost always causes obvious symptoms, i.e. pus dripping from the penis -- and the symptoms always begin within 2-5 days. So having no symptoms, you can be very sure you were not infected. Second, although 2.0 g is recommended if azithromycin is used for treatment, the 1.0 g dose would cure at least 90% of infections. Combining the low chance you had gonorrhea plus the effectiveness of the 1 g dose, all is well and you don't need additional therapy. (But if you decide to do it anywya, just take another 1 g dose. You still have lots of azithromycin in your system; a single dose gives a week of drug in the body.)
Regards-- HHH, MD