Plantar fasciitis, while felt at the foot, has quite a bit to do with core issues. Typically, the core is torqued because the myofascia in the pelvis and mid back distorts the core.
Since we base the way we feel and our "normal" on how we see ourselves at the age of about 6, we all have our bias based on what we have experienced. In other words, you can be very tight all the time, totally inflexible and think this is normal. The question should be is there a normal or is there a normal for you or does normal mean the way we always feel (sore, tight, etc) and then when we hurt, this is called abnormal.
The reason I bring this up is because we are all built differently, and to suggest any one type of shoe or way of walking is incorrect, or by switching from one shoe to the other created a problem, may be missing the point.
Since some of us are built more symmetrically than others, and since we come in all different shapes and sizes, it stands to reason that certain shoe designs may work better for you since it levels the hips and improves the function of the core which if working properly absorbs most of the force from the ground when running and walking. If running shod (regular shoes), this means a more open stride and more of a heel strike. If running barefoot, a mid foot strike is expected and according to those who advocate barefoot walking and running, more instrinsic foot muscle use and strengthening.
If the core functions poorly, no matter how you walk or run, you will pound the leg into the ground and end up with tight calves and other running issues such as plantar fasciitis.
Sometimes, changing the shoe style is more anatomically correct for your body style, alleviating the symptoms as the core works better and the legs becomes less tight. Other times, intervention may be necessary such as fascial release, art or graston technique.
One more thing - if the tissues surrounding the ankle and foot are tight, and do not allow the foot to move properly, it too will cause plantar fasciitis because the hip will also tighten in response, messing up your gait. Unfortunately, the knee often takes the brunt of this. Seeing a good sports chiropractor can help with the motion of the foot as well as the other articulations in the hips and back.