I have not heard of having the urge to pee as being a symptom of an STI. As most STI do not show symptoms I would recommend going for a test if you are concerned about an STI being passed to you.
The sex you described wouldn’t pass an STI to your urethra or urinary tract as it was covered by a condom. If you go to a clinic just describe the symptoms you are having and ask for an STI test. They would probably be able to do a urine dip stick in the clinic to see if there are any indicators of something being abnormal and then send the urine for STI testing. The health professional you see would also be able to send you for additional test if needed for example if they think the prostate may be responsible for the symptoms.
Let us know if you have any more questions or concerns.