You 'walk' a lot in the ring, crossing your feet almost every time you move. Once you get in the ring, you don't move that way except for very specific purposes.
You aren't doing yourself any favors fighting as a southpaw. About 90% of the punches you throw are left hands- you'd be better off having that activity from your lead hand. That lead hand has to be able to jab, hook, hook off the jab, double up on the hook, throw uppercuts, an uppercut then hook behind it, etc... 80% of what the rear hand does is defensive- catching punches, parrying, etc...-if you are doing this properly, and learning the punches would be a lot easier than trying to teach your right hand all that it needs to learn to be your lead hand.
In this era of boxing too many guys are trying to fight southpaw, thinking they'll get away with something- I think- and have an advantage in being able to land the left hand. Well, it isn't as easy as it looks and it does require a great deal of work with your right hand as well. In almost every case you are better with a strong lead hand than strong rear hand.