I'm a martial artist too!!
Mainly BJJ, but also some Muay Thai, boxing, wrestling and MMA.
So I would guess what you mean by right handed is you stand conventional, with you left leg forward right?
Because of course the power hand and power kick are the back hand and back foot, meaning if you are righty/conventional then your cross and uppercut in boxing will be thrown with your right hand (back hand) and your round house kicks mainly with your back leg (right leg).
I also am conventional and stand left leg forward, but I REALLY want to learn to stand south-paw (lefty with right foot forward) because it throws people off.
Anyways...I LOVE this topic because ambidexterity/mix handedness is a major interest of mine since I am basically ambi/mixed handed...not sure what the difference is.
Here's how I am:
I write and always have written almost exclusively with my left hand, but I do almost EVERYTHING else with my right hand.
I throw a ball with my right hand, as mentioned, throw my power punch in martial arts with right hand, pick most things up with my right hand, hold a fork in my right hand, everything except for writing I do righty, and everything else I try to do with my left hand feels just as foreign to me as writing with my right hand does.
However, I've become really into practicing writing with my right hand because it used to be so awkward for me and now that I've been doing it at least once a day most days for the past couple years I'm becoming MUCH more comfortable writing righty!!!
It's a cool feeling.
I haven't gotten into practicing doing many other things I am most comfortable doing righty with my left hand, but I am going to, and want to eventually become as close to purely ambidexterous as possible.
Because this post is already long, I am going to stop here and make a new one to continue this tangent because it excites me, and on other forums I've found that people get tired reading long posts and stop reading half way through, but if I make two average length posts they are more likely to read them both.