My coolant/temperature light has been continuously flashing on my Chev, even though my coolant reservoir is full; what gives?

Bad thermostat.

Thermostats are pretty easy to replace and pretty cheap. That would be a good first thing to replace.
Do you have any OBD codes to read?

What is the engiine and year of the vehicle . Are yo getting an OBDII code. P0128? You can go to Advance Auto and they will read the code for free.
You can have a bubble in the system or a bad T-Stat.

The temperature is getting too hot. Probably the thermostat. Not to hard to replace.

Could be an air bubble near the temp sensor. I had that happen to my Prius.

You know, it could be that the system is operating perfectly correctly, and you just don’t have sufficient antifreeze. Have a mechanic check the antifreeze protection, and then the thermostat.

Check your belts for tension, they may be loose or bad and your coolant pump may be slipping. Happened to me already where I got the same alarm as you.

If it’s not the sensor, what about flushing the transmission fluid? If the it’s crusty, would that cause the low coolant light to come on?