@jeffrey-white   I had a similar issue with my 37 with a newly rebuilt 235 and a new radiator.  It would blow coolant out the overflow tube and past the cap and drench the underside of the car with coolant.  It would continue to do this until the coolant level was so low that the engine temperature would start to increase (low on coolant).  I bought one of those combustion gas detector kits and tested the engine maybe a dozen times.  It never failed a test.  I finally purchased one of those after market overflow tank kits that mount on the firewall and I have never had another issue.  Radiator is always full, car runs consistently at about 160, even in 95 degree weather, and I have no coolant leaks.  I can't explain it but I did fix it.

@ole-olson I think you could add the early 4 cyl chevys to that list. And strangely my parts book for 29-32 list a thermostat and in bold letters advises NOT to use one with cars with a heater. Also I think the first hot water heaters from Chevrolet came out in 32. Maybe one of our cooling system experts will "chip"
 in and enlighten us.

Update:  I tightened the water pump packing nut just a little.   Also installed a temporary mechanical temp gauge so I could monitor the temp.  The original gauge is not working.   Took it for a 3-4 mile drive and the temp got up to 180 and stayed there while driving.   It did creep up to 220 after shutting it off which I’m guessing is normal when the circulation stops.   It did not puke any antifreeze after I shut it off this time.  There was some foaming in the radiator when I first started it up.  What is the normal operating temp for these engines?   

@jeffrey-white The normal temp is determined by the "thermostat" being used but I think the 180 is a little high . I run a 160 in my 32.

FWIW,  My 1929 LQ Truck 194 CI 6 cylinder, and using a 160 degree T-stat, I also experience coolant overflow at shutdown. But this only happens when I top off the coolant level per my every so often check of the internal check of the water level inside the radiator. 

I open the radiator cap and see no coolant water level inside.  The radiator internal cooling coil top is visible, but not the coolant level.  So I add my 50/50 mixture of coolant to raise the water level to be just at the top of the internal cooling coil top.  Note, upon my finishing adding the coolant, there is plenty of open area headroom in my radiator, say 2.5 inches or more, and when I take my truck out for a drive, the cooling system overflows out the excess coolant by way of the side overflow tube, and upon my parking the truck at a store or at home for the day, the coolant still leaks out.  Just for info, this coolant leaks out slowly due to it not being under pressure.

Also note, yes, the radiator cap must be left loose. Actually my cap cannot seal the radiator, which had it sealed the radiator, it would creating a pressurized system.  This cooling system is somewhat open to atmosphere pressure.

So, what I originally thought, per all my reading of the proper water level inside the radiator, per this 1929 design, did not hold true to my system.  The data is close though.  I look into my radiator every week and do not see the coolant level at all.  The level is below the top surface of the internal cooling coil body.  The coolant level is out of sight.  So, every other week I add my coolant to the top of the internal radiator coil.  Since I cannot see the true level my system wishes to hold, my every other week addition gives me a secure feeling that my cooling system water level "top-off" is proper. My 1929 cooling system will get rid of what is does not need.

And yes, every two weeks when I do this procedure, I have a large turkey roasting foil pan under the radiator to catch the excess coolant as it leaks out.

All this info is not to say is similar to your issue.  It may help others who think they are experiencing your issue but in fact may be experiencing what I have.

PS.  In the 60's and 70's when I was playing around with swapping engines, rebuilding heads and changing manifolds and carbs, we always put in a new thermostat.  There were times when those new thermostats did not work properly.  Those problem ones, most just locked closed. And we immediately new it was the T-stat.  But a few of those new ones, locked "partially open" which caused us headaches until someone pointed out the possibility of a bad t-stat.