My method is totally screwed up and weird but it works for me. I believe it is the result of self teaching myself how to fly fish and not taking any proper fly casting classes or having someone show me how to cast properly and this was way before the days of the internet and YouTube. Plus I am semi ambidextrous - I eat with my left hand but do everything else with my right hand, so I consider myself right hand dominate. I used to shoot my shotgun and rifles with my left but once I got a scope for my rifle, I found out I couldn't shut or wink my right eye, so I switched to my right and problem solved.
Anyway, I cast with my right hand and then instantaneously switch the rod to my left hand when done casting, so by default I also reel in with my right hand as well. I have been doing this for 30 years and it never been a problem nor do I ever recall losing a fish by doing this. Granted I primarily fish stillwaters so that makes it a bit easier and I can get away with it, but it hasn't been much of a problem when I am fishing moving water, but I do notice when fishing in fast water it does impact the drift for a second or two as I am making the switch, but I have never missed a fish when one instantly hits the dry fly when it hits the water, actually it can help as there is a bit of pause allowing the fish to take the fly without me setting and pulling the fly out too early.
What makes a bit more odd is that it took me years to even know that I was doing it. I finally recognized my odd style when I was trying to help my brother-in-law and nephew set up their rods and reels over text/phone calls. It caused alot of confusion when I was explaining that I casted and reeled in with my right hand. They finally called and asked what the Hell? I had to go through the motions and the light bulb went off that I was doing it this odd ball way

. Anyway it works for me, and I am too set in my ways to change.
So yes, reeling with your right hand is OK

.
Peach