Is the continuous dripping of urine after performing istinja and wudu, and the feeling that one's prayer has become invalid because of it, considered a waswas (obsessive whispering) or an illness?

Worrying (waswasa) is one of the most dangerous ailments, and its cure is to ignore it and pay no attention to it. If you are not absolutely certain that urine has exited, then the default assumption is that it has not. So, abandon doubt and do not give it any importance. If you are certain that drops of urine have exited, then wash the area affected by the urine and perform ablution (wudu’). There is no need to change clothes. The Malikis hold that it is not obligatory to purify the body and clothing from urine if it exited involuntarily. If the emission of urine is continuous and you do not find time for purification and prayer, then perform ablution after the time for prayer has begun and pray the obligatory and supererogatory prayers with this ablution, because you are afflicted with سلس (urinary incontinence).