#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. Since the set theoretic question may require different ideas perhaps I will not give the reference (in any case it was not by me).

#Pie-in-the-sky question

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

#Actual question
In lieu of an answer to (I) I could mark as "accepted" an answer to one or both of the two natural related questions:

II. Is it possible to add an eventually different function to $M$ without adding a dominating function?

III. Is it possible to add an eventually different function to $M$ without adding a Solovay real?