All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad, sallallaahu 'alayhi wa sallam, is His slave and messenger.
If what you are experiencing is merely Waswasah (obsessive whispering), then you should ignore it and pay no attention to it. But if you are certain of what you describe, then you should take precautions by placing a piece of cloth or something similar on the appropriate spot, so that impurity does not spread to your clothes. Once you are sure that the dripping of urine has stopped, you should remove the cloth, clean yourself, perform Wudu (ablution), and then perform prayer provided that, within the time in which the obligatory prayer must be performed, you can find enough time to perform the prayer in a state of proper ritual purity, which seems to be your case.
However, if within the time in which the obligatory prayer must be performed, you truly cannot find enough time to perform the prayer with full ritual purity (i.e., free from dripping), then your case is like that of someone who suffers from urinary incontinence. In this case, you should make Wudu after the prayer time begins, and with that Wudu you can perform the obligatory prayer as well as however many voluntary prayers you want to pray until the time of that prayer ends.
Allah knows best.