If they did a urinary catheter during your surgery, the biggest risk factor is infection. Damage to the structures during catheterization is very rare; possible, but rare.
What you're describing sounds like 2 things/2 different symptoms: Urinary urgency and Urinary retention (the not feeling like you're emptying, then having to "go" more). You should see a doctor for both. Get checked for a urinary tract infection. Both can be symptoms of a urinary tract infection (among other things).
My experience: They did not insert a urinary catheter for my surgery (I know, I asked). But, within about a month after surgery, my urinary issues were better. Before surgery, I dribbled and would easily wet myself if I coughed, sneezed, or laughed too hard. No amount of Kegels seemed to help. After surgery, I no longer dribbled and I these days it's pretty rare for me to have issues with coughing/sneezing/laughing.