Sounds like a good problem to have!
I have a similar setup (10 gallon batches, kegerator holds 5 kegs, I have around 10 kegs total), and here’s what I generally do:
If kegerator has space, newly kegged beer goes directly in and gets put on carb. Cold crash is optional.
If kegerator is full, I generally try to leave in fermentors for as long as possible. However If I need fermenting space (or if I am really anxious to get a beer on tap), I will rack to a keg, add priming sugar, and seal it with a blast of 30 psi. It then sits for 2-3 weeks at room temperature to allow the beer to naturally carb up. If it’s a beer that I know I will be putting on tap soon I dry hop during room temp carb period if recipe dictates.
There’s no problem letting your beer sit at room temp in the keg. And adding priming sugar to let the beer naturally carb not only ensures your beer will be ready by the time it goes into the kegerator, but also saves CO2 and I believe allows the yeasties to munch and clean up the beer a little more. It also helps to limit the amount of O2 exposure and infection risk.
If kegerator is full, room temp kegs are full, fermentors are full, and I still want to brew some beer soon…. I have a party! Or I bottle up some of the kegs that aren’t getting as much love (that 2 gallons of whisky barrel double imperial stout that’s been taking up fridge space for 6 months can be bottled). That scenario has only happened maybe once in my brewing career.