I don't understand why you would be getting D# and E, because B is nowhere near that pitch. Are you using a tuner designed only for bass guitar? There isn't a B string on a four string bass, so a tuner set for bass guitar wouldn't pick up the correct pitch for a B string.

I don't think it's a problem with the string sticking at the nut, as that wouldn't stop you getting closer to the pitch.

Try watching some videos on how to tune a guitar