Embarassing question about labiaplasty; I have sui. What will catch it if I dribble?

Labiaplasty. Reducing the labia will not affect your sui. If you leak you will still need to wear a pad if your leaking is significant. By the way, your sui can be easily treated with a simple, safe, and effective procedure called a mid-urethral sling.
Not afterward. It should not affect your labiaplasty should you choose to have one. :).
Cleanse & hygiene. In fact, many patients state that their large labia retain urine from interference with the urinary stream as well as difficulty with maintaining constant hygiene due to folds and creases. This has been reported as one of the benefits of surgery ot reduce retention of urine. Treatment of the sui is an entirely different issue and should be addressed with biofeedback, medication and/or surgery.