What is your CPU utilization like when this happens? If it's less than 100% of one CPU thread, then the most likely answer is that you're not actually overloaded but you're doing something to block the event loop. Set the environment variable PYTHONASYNCIODEBUG=1 and see if that points out any blocking operations.
If your CPU reaches 100% of one CPU thread, then you are actually reaching the capacity of one python process and need to run more. Note that you must run more *processes*, not just more threads - the python global interpreter lock (GIL) means that adding more threads to a python program won't help you use more CPUs effectively. This traditionally means introducing a load balancer to direct requests across your multiple processes, although the relatively recent (erm, 2013, so not so recent now) introduction of the SO_REUSEPORT option should simplify this (assuming your multi-process workload fits within one machine).