You can still get herpes, syphilis and HPV, but those wouldn't give you discharge or itching in your urethra.
If you have to squeeze your penis to get a discharge, it's probably not a discharge related to an STD. A discharge from an STD will be there without you squeezing.
If the itching doesn't get better, you should get checked. You could be starting a UTI, which isn't related to sex. If you've had any unprotected encounters (either oral, vaginal or both), you should get checked for STDs.