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 is His slave and Messenger.
If your feeling that you discharge these drops is just an illusion or an obsessive doubt, then do not pay any attention to it. You only need to wash your private parts without taking the doubts that you may have afterwards into consideration.
However, if you are certain that you discharge these drops of urine, then you can place a napkin or the like in that place and wait until you are sure that these drops stop, and then you throw away the napkin and wash your private parts without the need to wash your clothes. Thus, you will not feel embarrassed, as you mentioned.
If you do not do this and you discharge urine, then you must wash the clothes from this urine, even if this leads to what you mentioned, as the right of Allah is more deserving of being fulfilled, and you should be embarrassed before Allah rather than before people.
Allah knows best.