"Check Hybrid System" after engine swap, no A/C but runs normally

I think I've checked every thread on this subject. Please be patient with me, I don't have much in the way of money or resources.

I live in a rural town. The closest Autozone or Toyota dealership is 45 miles away. My friend has an OBD2 scanner that I can access tomorrow, but I don't know if it can read hybrid codes. Also, I live in this car.

The events: 2013 Prius with 189,880 miles went ~7,000 between an oil change. The oil and filter had silver and bronze colored flakes in it. My friends and I got a donor engine (2013 with ~100,000 miles) and swapped it in. It started up fine.

Current problems:
- The dash shows, "Check hybrid system, stop the vehicle in a safe place" and the yellow triangle. Otherwise it runs entirely normally.
- Also, the air conditioning doesn't cool air when I press the button even though the button turns green. The fan sounds louder when the button is green.

Already considered:
- There is plenty of (new) engine coolant and no signs of leaking/depletion.
- The inverter coolant is also full and getting circulated. The inverter wasn't touched during the engine swap.
- Inverter coolant and all other fluids were changed at ~182,000 miles.
- None of the A/C was removed during the engine swap.

Relevant notes:
- The engine coolant temperature sensor wires got pulled out of the black plug (the sensor directly behind the large coolant line on the engine's driver side). My friend put the wires back into the plug. I thought they got shorted, so I removed them and wrapped electrical tape around one to separate them, and pushed them back in. They seem kind of loose in that plug.
- The A/C plug under the hood had coolant dripped on it. I tried to clean both ends out with a rag with the battery disconnected, but upon restarting, the problem was not fixed.
- The Energy Monitor screen sometimes shows no arrows when the engine or electric motor are in fact running.

Questions:
- Ideas for things I can check? How to fix both of the plugs?
- Is it safe to drive 45 miles to the dealership in this condition, if my friend's scanner doesn't pick up hybrid codes?

 

Update:
I got codes P0351, P0352, P0353, and P0354 - ignition coils A-D, primary/secondary circuit.

 

Is there something I'm not including that would get a response?

The engine temperature light also started flashing when I drove <10 miles at highway speed, or 3 miles up a hill at 20 mph.

The cabin heater still works.

I just added more coolant, since it went from "just below full" to "just below low" while sitting for 2 days. There is no evidence of leaking. I bled it by pouring coolant into the reservoir (to B level) while the upper coolant temperature sensor was removed. Then I put the sensor and cap back on, and I ran the car in maintenance mode for 10 minutes with the heat on high. The coolant level is still at B.

Question: Could the Check Hybrid System warning be related to the engine overheating?