This is why I deny frequency change to those departures whose first order of business after handoff from tower is to request a freq change.
Approach controller should recognize the conflict and vector either one or both of you to avoid. Obviously there will be situations where their attention may be elsewhere, in which case, I wouldn’t begrudge you altering course momentarily to avoid.
Unfortunately, many times when I try to vector the second plane (if they’re departing traffic), it takes several tries and too much time to get past the Unables [which, for those that are unaware, does not mean “I don’t wanna”], because instead of looking at their map or just, you know, following instructions, they don’t want to adjust from their single waypoint flight plan for even a couple seconds. Again, there, you are not begrudged an evasive maneuver.
In the event that they are climbing, departing traffic and you descending, arriving traffic, most of the time the separation is most expeditiously achieved by a slight acceleration of your descent, passing below them as they climb out, though not always the case, obviously.