Darse de navajazos - does this mean to have a

darse de navajazos - does this mean to have a fight?

That seems to me a correct idea … and a fight with knives!

Yes, a fight with knives.
navaja = folding knife
navajazo = a strike with such a knife
in general, the “azo” suffix is often used to mean a strike with something: portazo, puñetazo, porrazo, batacazo, …

darse de = exchange [blows, …], hit each other. It’s the reciprocal equivalent of “dar un golpe, palo, navajazo, …”

darse de palos = exchange blows with sticks
darse de navajazos = exchange knife cuts
darse de golpes, de puñetazos, de hostias [slang for blow], …

Gracias! Very informative.