Not crazy as Robert has said. Yes, get rid of the old stuff and match all new nozzles.
You will have to do more digging though, if you install the rotary nozzles, because they require a spray head. I20 rotors are 3/4" inlet vs. 1/2" inlet for a spray head. If installing spray heads, you will have to dig all the way down to the fitting to see if the thread is 3/4" or 1/2". If 1/2", then install the right size nipple from fitting to the spray head. If 3/4", then install a 3/4" to 1/2" nipple to the spray head.
Your rotor zones could be a little stretched causing the dead spots from lack of water. The rotors could have been diffused too much which shortens the nozzle stream. They could also have the wrong nozzles installed.
If it was me, I would replace the rotors, because the retrofit would be easier. Then, install the correct nozzles to give you the overlap you need.
I personally use Rain Bird PRS+ rotors. I love the nozzle stream coverage from rotor to end of stream. The rotor is also pressure regulated to 40psi and has a little valve on top that lets you turn the rotor stream on and off if you want to change nozzles.
I'm also curious to your yard dimensions. I'd also like to know what the head lay out is and the distance between rotors.
Looks like a fun project you're getting into. Good luck and have fun learning!