#Definitions I believe set theorists have studied all of the following three notions in the context of forcing extensions of a model of ZFC, $M$ (hopefully the terminology is the standard one).

1. A function $f:\mathbb N\rightarrow\mathbb N$ is eventually different if for each function $g:\mathbb N\rightarrow\mathbb N$, $g\in M$, the set $\{n: f(n)=g(n)\}$ is finite.

2. A real $r\in [0,1]$ is a Solovay random real if for each measure-zero subset $S$ of $\mathbb R$ with $S\in M$, we have $r\not\in S$.

3. A function $f:\mathbb N\rightarrow\mathbb N$ is dominating if for each function $g:\mathbb N\rightarrow\mathbb N$ in the ground model $M$, the set $\{n: f(n)\le g(n)\}$ is finite.
   #Motivation
   An eventually different function that is not too fast-growing is reminiscent of a random real. Can we always use it to construct a random real? The analogous problem in computability theory was quite difficult but has been solved by Kumabe and Lewis (J. LMS, 2009).

#Question

I. Is it possible to add an eventually different function to $M$ while adding neither a Solovay real nor a dominating function?

EDIT: Now stating the question in the strongest possible form, which is the one Andrés Caicedo answers below, and revealing the reference.