'03 Tahoe Overheating

['68OrangeSunshine]

Monday, I took the 2003 Tahoe to work -- about a 34 mile round trip. Morning run included about ten miles on the freeway. [I have alignment or tire issues but that's another gripe for later.] Morning run was good, and the A/C surprised me by blowing cool. I had thought it was not working.
On the return run, I don't have to use the freeway, I can run on streets and boulevards. Since I had to work the next morning, early, I stopped at the In-N-Out for a double-double to go. With the A/C on, I taxied thru the multi-car line on a 104* day. [Still new at this.]
On the run home, there's a big hill. As I got up the hill the truck went nuts with the Goat Bell, and the scroller read ENGINE OVERHEATING. I pulled into a Chevron station, popped the hood and watched orange coolant violently bubble in the overflow tank. I let the turbulence subside and cracked open the filler cap, by degrees. When most of the steam had bled off eventually, I pulled the cap. A small orange geyser chased my gloved hand, but missed me. I had not included any antifreeze jugs of water in the Tahoe. I filled an empty wiper juice bottle from the coin-operated water machine at the station. Fifty cents a gallon. Took two gallons.
I got the 3 more miles home OK.

I am used to putting a T-Fitting from a garden hose into the heater hose line to flush my engine block, heater core, and radiator on my '68 and '71 trucks.
Not sure how to flush this system. The Radiator cap seems to be on the puke tank.
I bought one gallon of Autozone house brand orange coolant. Probably needs 2, and a couple gallons of distilled water.
How do I know if the Radiator needs replacing?