i think yr problem might be having a fast let down. milk overflowing , as what you claim yr baby dripping milk out fr mouth. flow is too fast for baby to swallow all milk.
solution: you can pump out some milk first before latching her direct. normally it is the first let down that is too fast.
the direct breast leaking milk while feeding. if you are using/wearing nursing bra, it will be easier. you can also put the breastpad in bra to absorb the milk if you are wearig nursing bra or those cotton bra without wire. i believe very messy only if you are totally bare while feeding her.
you can try to prop baby up with pillow pn yr arms or back to create most comfy position to feed. can also use breastfeeding pillow.