Rather than guessing about the condition of your '36 radiator I would suggest that you take it to a radiator shop and have it inspected, cleaned and tested.
I know that radiator shops that actually work on radiators are quite rare, the good ones usually specialize in truck radiators, however, a radiator is a radiator.
I have had excellent success using aluminum radiators from SpeedWay/Summit that are made for stock car racing. We use cross-flow radiators, standing them up. By using a radiator for a GM in a Ford the in and out tubes are in the correct position. Of course for a flat head, additional upper and lower tubes have to be added. The fill neck being on the side of the upper tank is an added bonus.
The aluminum racing radiators have large tubes and thick cores so they really cool an engine.