You are having a special case of a Ornstein-Uhlenbeck process; in your case the mean-reversion part is stable.
In general, an OU-model can be expressed as:
$dX(t) = \alpha[\theta - X(t)]dt + \sigma dB(t)$
- $\theta$ being the process' optimum / "mean you are trying to revert back to"
- $\sigma$ being the intensity of the process' random fluctuations
- $\alpha$ being the strength of the selection towards the process' optimum $\theta$
As you see, the first term that you call "drift", is practically the "strength of mean reversion" towards $\theta$ and usually get larger the further away you are at time $t$ from $\theta$. In your example this term is fixed making $\theta$ and $\alpha$ are non-identifiable but still to call your increments white noise you need to subtract the "(deterministic) drift" $d$.
So yeah, your intuition is right, $d$+ $\epsilon_t$ is not white noise, if $d=0$ isn't true. (In which case you have Wiener process)
I am getting to all the trouble to say this stuff cause the OU process in the only stationary Gauss-Markov stochastic process out there so if you are inclined you can really do fancy stuff stochastic-wise with it. :-D