Is it necessary to wash the private parts and underwear for every prayer, given the presence of gas, traces of moisture, and urinary incontinence, knowing that purifying the private parts takes a long time?

What you are doing is a type of waswasah (obsessive whispering/doubts). One is not required to perform istinja (definition)' (cleaning oneself after relieving oneself) due to wind unless there is absolute certainty that something necessitating it has exited. As for urine, it is sufficient to pour water over the affected area until it is highly probable that the impurity has been removed; there is no need to wash areas not touched by the impurity. If you suffer from incontinence (sils), then perform istinja' after the time for prayer begins, take precautions by placing a cloth, make ablution (wudu'), and pray as much as you wish until the prayer time expires. There is no need to renew the precaution for every prayer unless you intentionally invalidate your ablution. The Maliki school offers تسهيل (facilitation) for those afflicted with incontinence, and their opinion can be followed when necessary.