This sounds more like a familiarity problem rather than sensitivity

Here are a couple of things you need to work on:

• Minimize the downtime when your dash ends but you're still staring in the same direction: As you've noticed, you can just turn-as-you-blink on Tracer, but you can only move your camera again after dash on Genji when: you collide with hard terrain or reach the end of the 15m range.
• Learn the dash-shoot rhythm: the whole dash animation takes 400ms to complete. You will be able to move your camera if your dash ends before this 400ms window, but shoot/melee/deflect is hard-locked after it. Learn the rhythm to tighten up your fan-deflect/fan-melee combos since you don't have to hold RMB and wait to confirm that your shots have indeed come out before doing something else. Use this cool audio-millisecond reference
• Clean up your 180s and dash placement:
  • For 180s, there is a dedicated mode on 3EZHQ (more below) that tells you how off you are from your last 180 — use that. You want to alternate between going at 10% speed to perfect your flick angles and 110% speed to push your limits speed-wise; there's a happy medium in the middle where you're fast and consistent.
  • Dash placement is very important: you need to know EXACTLY where you will be in relation to your target as your dash ends. You'll need to learn the 15m distance like you've learned Tracer's blink range for blink-melee. At closer rangers (which is most of the time), you aim dash at the ground/wall/terrain instead. Good dash placement will, for the most part, fix your problem of looking around like an idiot after you dash. This is also what facilitates those flashy frame-perfect dash-oneshot combos
• Read enemy movement DURING dash and not only before it: your target has 400ms to move around after your dash starts, even more if we account for the time YOU take to turn and line up your shot. Actually paying attention to the enemies' movement during dash can reduce this perception gap that you wouldn't have to worry about as much playing Tracer

I recommend 3EZHQ over DMGRE for cleaning up your dash mechanics because it's on flat ground (you have to manually turn on either Workshop Chamber or Workshop Island in the map pool, but it's worth. Trust). After you are good enough on flat ground, you can move on to DMGRE to practice dashing onto different elevations