Can you change airways when they cross without a named fix?
Yes one can fly one airway to intercept another. Many reasons exist as to why not all intersecting airways have an intersection, the simplest being, it wasn't necessary.
I will suspect one may encounter difficulties in executing your desire to change airways as stated. Such would be the inability to file a flight plan with dispatch or that flight planning software can't/won't accept this unconventional navigation method and the most illogical, if one could accomplish the later, I would image would be ATC. Why? Because the controller has never encountered this situation, before and doesn't want to explain their actions should something go afoul.
For years ATC would issue preferential routes to aircraft departing KPIA to KORD as such; PIA PIA030R V10 PLANO ORD. Aircraft would depart KPIA on the tower assigned departure heading then be vectored to join the PIA 030-radial to intercept Victor-10 northeast of BDF VOR onward to KORD. Then computers became more involved along with flight management systems and pilots were no longer capable to navigate this route because computers couldn't calculate the routing or how to intercept one radial (airway) from another without a set point (intersection) along the second radial , something pilots had done since the 1950's.