It sounds like corroded wheel cylinders which are causing one side to stick. It could also have collapsed brake hoses. The usual senario is that one side sticks and the other side locks up. You don't have to switch to disc brakes as C-30 drum brakes are totally adequate when working properly.
Fred


1956 3100 Pickup/Red/350/3sp OD/PS/Disc Brakes
1957 Bel Air Sport Coupe/Red/355/TH350/PS/Disc Brakes
2017 Silverado LT Single Cab SB/Black/5.3/6 Speed Trans
1947 Willys CJ2A w/F-Head engine