I play all my cajun and tex-mex and Chris's Frisco Zydeco on my single row. I only play a single row. Chris was very encouraging and yes it can be done. My Martin is wet tuned 5 cents and I close #1 and #2 reed banks. You get a pretty authenic sound. If you dig a little you will find that the single row was used for conjunto in the beginning but soon switched to the double row. You can still find a few old timers like Wally Gonzales who play some conjunto on a single row. He also plays a mean toy accordion.
Two straps may not look "cajun" but if its any comfort the accordion is not "cajun". We "adopted" it back a few years ago. Like everyone says, nobody cares what it LOOKS like, its the sound that gives us the warm fuzzies.
I play with two straps(one over each shoulder) and center my thumb behind the keyboard. Been playing like this since 85 and I'm passing a good time.