The Engine Serial number T5978813 is good for a 1936 Chevrolet Truck. My book shows it was built in January 1936. I don't know what the 1T is for, but my guess is you have a 1-ton special ordered truck. Chevrolet on paper only built Series "FB" 1/2 ton Commercial and Series "RA", "RB", "RC" & "RD" 1-1/2 ton Trucks. They would down size the "R" series trucks to 1-ton rating for "special orders".
1937 was the first year the 1-ton trucks got their own separate Series identification.
1946 was the first year Chevrolet made a 2-ton truck.
Your body number doesn't look right. It should look like this... 3RA01 1234
3 = plant (St. Louis)
RA = Year/Series (1936 1-1/2 ton Single Wheels)
01 = Month (January)
1234 = Unit Number starting at 1001
536CI ??? Where did you find this number?
The weights you've given again lead me to believe it is in fact a special order 1-ton truck.
What type of body does it have?
Does it have mechanical brakes or hydraulic? The very early '36s still had mechanical...
Post a picture so we can see it!