From my personal experience, for the battery - I would be testing my alternator/generator. It must not be charging the battery as you drive.
As for the radiator - I was recommended a 4-7lb cap. Replace it and run the truck for a bit. See if thats where your antifreeze is coming from. Is your thermostat working like it should? On another project, I had bought a cheapy one and it didn't open as it should. Had to replace it.
I just got done replacing the radiator, thermostat, belts and hoses on my 53 so once I had it all together and filled it up with water/antifreeze, I let it run to make sure the thermostat was going to open at 180 degrees like it should. I watched with the radiator cap off and sure enough, 180 degrees and it sucked the water right in like it should. Take a look and see that your truck is doing that.
