360 running hot…very hot

IMO, this is real simple;
If your rad is dropping the temp 30degrees, from top to bottom, then your cooling system is working exactly right. Go to Note-5
If only 20*, it could be better but as soon as the car is moving even just 10 mph, the temp should be dropping. There are several things you can do to make it better, most of which have already been mentioned.
BUT
the first thing to do is make sure that your Idle/slow-speed timing is NOT retarded. See note-1
This puts heat into the cylinder walls and into the exhaust ports; where it is picked up by the circulating water.
The temperature of the headers within the first four inches of the flanges, at idle should be less than 450*F. see Note-2
After you get all that bugged out, and you still cannot get to a 25/30* temperature drop across the rad at idle with the vehicle stopped, and if this is a recent engine-build, then see note-5

Happy HotRodding

Note-1
Your Idle timing, with an automatic and the stock 360 2bbl cam should be around 8>10*. The more cam you put into her, the more Idle-timing it will want. Some of the FABO guys are running 18 to 25 degrees. But the more timing you run, the more closed the transfer slots in the carb will be, which leads to a throttle tip-in sag, and she may bang on the N/P to in-gear shift. The mechanical timing should begin advancing about 100>200 rpm above idle, and should hit 25 to 28 degrees by 2800rpm, and then slowly creep all-in by 3400 +/- 200
Your vacuum advance MUST be working.

Note-2
Closer to 400 or even 350 is better. If it's hotter than 450, I would suspect that the exhaust gasses are exiting the chambers still on fire.
Here's an easy test; just pull some timing in, without regard to how much, then re-adjust the idle speed back to "normal". Just keep repeating this until adding timing produces no additional rpm. Now read the header temperatures, and watch the temp gauge. Put the timing back after the test. See Note-3

Note-3
Caveat; if the carb's low-speed circuit is exceptionally rich, and you have a long overlap-period cam, and the headers are working really well, then the headers at too slow an idle can draw A/F mixture right across the top of the pistons and out the exhaust ports, and into the headers. Under the right circumstances, this raw mixture can ignite and artificially increase the header temperature. see Note-4

Note 4
If you have a metering-rod carb, make sure the rods are staying DOWN at Idle. If a PV you shouldn't have trouble unless the diaphragm is ruptured.

Note-5
Ima betting your ring-gaps are too tight and or, not enough skirt clearance.
Here's the easy test; Make sure your battery is in good shape. Then run the temp up to ~210 by IR gun, then shut her off. Wait 5 minutes than crank her up. If she fails to crank, or cranks real slow/with much difficulty, STOP! the pistons or the rings have locked up, and the engine needs to come apart.

Note-6
If the engine is hopped up, it behooves you to, at your earliest convenience, to rig up a fresh cold-air intake system. Hot underhood air is less dense and will appear to the carb as being at altitude. Wiki says that you are at 259 feet. Your engine can make good power there on fresh air.

Note-7
I should mention that;
with your type of shroud,
at idle and vehicle stopped,
the fan can actually draw hot air backwards at the corners, from under the hood to around the corner, and back thru the rad. The paper-test will prove it. Once the car gets moving, ram air will prevent that.

Note-8
forget about electric fans at least until you get your heat-issues solved.
Once you do get your heating solved, if it was me, I would just install a Thermostatic clutch on the biggest fan I can find, installed into a full-coverage shroud. Let the clutch take care of business.
Oh wait,
I already did that, with killer results!
In your case, I would also recommend a hi-flow pump which is just an 8-vane pump with an anti-cavitation plate installed, and usually, a HD bearing to support that big-boy fan/clutch. I have had excellent results with a Milodon.