Overheating

Welcome back to 2CarPros. From your description it sounds like you may have got it hot enough to blow a head gasket(s). That would account for the barely running and sputtering. To check if this is the case you can do a couple tests, one would be to top off the radiator on a cold engine. Now start the engine, if the coolant overflows or starts to bubble a lot it is most likely a gasket. You can confirm that by using a combustion gas test kit. You can usually get them as a loaner tool from many parts stores. https://www.2carpros.com/articles/head-gasket-blown-test

The reason the 1965-1968 Mustang/Falcons have an issue is the radiator is flawed. If you look at them the inlet and outlet hoses are on the same side of the radiator. What that does is basically turn that nice wide radiator into a narrow one. The coolant takes the easiest way through so it comes into the top of the radiator and then drops down through the tubes. On many of them you can feel the difference, the hose side will be hot while the opposite side is barely warm.
There is a somewhat easy way to cure the problem. You install the radiator from a 1969 and up Mustang, it will have the staggered hoses instead of the stacked. To connect it there are crossover hose kits that replace the lower hose and connect to the new radiator outlet. That is the first image. They are getting hard to find, however it wouldn't be hard for an exhaust shop to bend you up one or even have one welded up out of stainless. The other method, which could be easier as you will probably be replacing the head gaskets would be to replace the front timing cover with a cover from a 1969 and up 289, then install the 1969 water pump. It has the hose on the opposite side already.
Now on top of this is another detail. Modern gas that has ethanol will burn leaner and therefore hotter than the fuel that engine was designed for. The only cure for that is to richen the mix up a bit or run only non-ethanol gas. Both work. Also check that the distributor isn't advanced too far. That will also cause issues with overheating.
Finally be sure you have the fan shroud installed, without it the fan doesn't draw the heat away very well.
Good luck and great to hear a classic is still on the road.