You're not doing it wrong, cables corrode/wear out. If the bike is that old and rusty, replacing the cables is normal, as is the tires/tubes, the brake pads, spokes (they might snap if you're truing up the wheels), chain links (if any are corroded), basically all of the wearable non-frame stuff. If there is rust on the frame, it might be structurally compromised and unsafe to ride.
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