Does it mean that I need to go to the bathroom?
Nope, bathroom here is just the actual bathroom itself. There is no toilet in mind.
The need for a toilet specifically requiring "nhà ve sinh"?
It can be: nhà vệ sinh, phòng vệ sinh, vệ sinh (in colloquial speech) or just a "toilet" with either a downward or upward inflection (toi-lét, toi-lẹt, toa-lét, toa-lẹt)