I'm not sure you are doing anything wrong, except that you may be mistaking pain during movement for something causative. While you feel the pain during your walks, it could well be the accumulated damage caused by your runs. Walking simply sustains the contractions longer, hence the pain.
Not always, but often, muscles stiffen up to protect themselves from further damage due to overuse. They sometimes lose this stiffness, after exercise has warmed them up and increased circulation. This is one reason why a stretch before exercise is less safe, since the muscle may pull back against the stretch, compounding the stress and injury. The potential for harm decreases as the muscle warms and relaxes.
What I often encourage among athletes, is to develop some skill for massaging stiff muscles when they are not cooperating during exercise. Just mobilizing the tissue without placing it under direct stress, can encourage peripheral circulation and lymphatic flow in a way that nourishes without jeopardizing recovery. I think this was the original idea behind stretching, which affects muscle problems somewhat, but indirectly.
Often times, there are only a few problem fibers causing the entire muscle to lock up. It can take some practice, but these small areas of spasm can be located with the fingers and deactivated with direct pressure. In other words, the elasticity of your muscle is not something you get by assuming it is relaxed, so much as by directly relaxing it.
I'm willing to bet there is something more to be known about your body mechanics, that tends to induce the tight and painful condition you experience. Sometimes, hip and foot structure play a role in how leg muscles are used during movement, leading to what we call overuse injuries. An example can be what happens to those who overpronate the foot during ambulation. It may not be that you are doing anything "wrong," but simply responding biomechanically to your unique anatomy. That response may require a lot more work from your muscles than it might on another person's frame. In such a case, modifications to footwear can help, along with some therapy to strengthen those areas that are being asked to work harder than usual.
Consider modifying your daily routine to give the painful calf muscles some rest, followed by some activity that focuses on strengthening your calves. Before this will work effectively, you must examine those muscles for tight bands of tissue that prevent the muscle from acting normally, and work out the kinks that are standing between you and success. Various techniques shown below.