Achievement of continence is problematic with exstrophy, as none of the normal structures to control the bladder exist, and they must be created through surgical intervention. Some techniques include vesicostomies, Mitrofanoff diversions, augmentation of bladder volume with sections of sigmoid colon, and so forth. In some cases, normal voiding through the penile urethra can occur, but in others a diversion of the urinary flow through means of clean intermittent catheterization through a stoma, or surgically created port, is necessary. Artificial bladder neck devices to control the bladder are available. Some individuals with epispadias experience incontinence, but others do not (see https://www.urologyhealth.org/urologic-conditions/epispadias).
By the way, people with hypospadias can have difficulty with incontinence, too, although it is often more a leaking situation rather than a reduction in bladder voiding control. Lots of other conditions can cause incontinence, such as spinal cord damage or weakness of the pelvic floor muscles. Incontinence issues can be difficult to handle emotionally, but various treatments are available.
Note: Incontinence pads and briefs for men and women are available (you can buy them online if you’re shy).